Chapter 13 Resource Masters - Math Problem Solving©Glencoe/McGraw-Hill iv Glencoe Geometry...

Chapter 13Resource Masters

Geometry

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3

ANSWERS FOR WORKBOOKS The answers for Chapter 13 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-860190-8 GeometryChapter 13 Resource Masters

1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03

© Glencoe/McGraw-Hill iii Glencoe Geometry

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 13-1Study Guide and Intervention . . . . . . . . 723–724Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 725Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 726Reading to Learn Mathematics . . . . . . . . . . 727Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 728





Chapter 13 AssessmentChapter 13 Test, Form 1 . . . . . . . . . . . 753–754Chapter 13 Test, Form 2A . . . . . . . . . . 755–756Chapter 13 Test, Form 2B . . . . . . . . . . 757–758Chapter 13 Test, Form 2C . . . . . . . . . . 759–760Chapter 13 Test, Form 2D . . . . . . . . . . 761–762Chapter 13 Test, Form 3 . . . . . . . . . . . 763–764Chapter 13 Open-Ended Assessment . . . . . 765Chapter 13 Vocabulary Test/Review . . . . . . 766Chapter 13 Quizzes 1 & 2 . . . . . . . . . . . . . . 767Chapter 13 Quizzes 3 & 4 . . . . . . . . . . . . . . 768Chapter 13 Mid-Chapter Test . . . . . . . . . . . . 769Chapter 13 Cumulative Review . . . . . . . . . . 770Chapter 13 Standardized Test Practice 771–772Unit 4 Test/Review (Ch. 4–7) . . . . . . . . 773–774Second Semester Test (Ch. 8–13) . . . . 775–778Final Test . . . . . . . . . . . . . . . . . . . . . . 779–784

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 13 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 13 Resource Masters includes the core materialsneeded for Chapter 13. These materials include worksheets, extensions, andassessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 13-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Geometry

Assessment OptionsThe assessment masters in the Chapter 13Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 724–725. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

1313

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 13. As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Geometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

congruent solids

ordered triple

similar solids

volume

Study Guide and InterventionVolumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1

© Glencoe/McGraw-Hill 723 Glencoe Geometry

Less

on

13-

1

Volumes of Prisms The measure of the amount of space that a three-dimensional figure encloses is the volume of the figure. Volume is measured in units such as cubic feet, cubic yards, or cubic meters. One cubic unit is the volume of a cube that measures one unit on each edge.

27 cubic feet � 1 cubic yard

Volume If a prism has a volume of V cubic units, a height of h units, of a Prism and each base has an area of B square units, then V � Bh.

cubic foot cubic yard

Find the volume of the prism.

V � Bh Formula for volume

� (7)(3)(4) B � (7)(3), h � 4

� 84 Multiply.

The volume of the prism is 84 cubiccentimeters.

7 cm3 cm

4 cm

Find the volume of theprism if the area of each base is 6.3square feet.

V � Bh Formula for volume

� (6.3)(3.5) B � 6.3, h � 3.5

� 22.05 Multiply.

The volume is 22.05 cubic feet.

3.5 ft

base

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the volume of each prism. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

7 yd4 yd

3 yd

4 cm

6 cm

2 cm

1.5 cm

10 ft15 ft

12 ft

30�15 ft

12 ft

3 cm

4 cm

1.5 cm

8 ft

8 ft

8 ft


Volumes of Cylinders The volume of a cylinder is the product of the height and the area of the base. The base of a cylinder is a circle, so the area of the base is �r2.

Volume of If a cylinder has a volume of V cubic units, a height of h units, a Cylinder and the bases have radii of r units, then V � �r 2h.

r

h

Study Guide and Intervention (continued)

Volumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1

Find the volume of the cylinder.

V � �r2h Volume of a cylinder

� �(3)2(4) r � 3, h � 4

� 113.1 Simplify.

The volume is about 113.1 cubiccentimeters.

4 cm

3 cm

Find the area of the oblique cylinder.

The radius of each base is 4 inches, so the area ofthe base is 16� in2. Use the Pythagorean Theoremto find the height of the cylinder.

h2 � 52 � 132 Pythagorean Theorem

h2 � 144 Simplify.

h � 12 Take the square root of each side.

V � �r2h Volume of a cylinder

� �(4)2(12) r � 4, h �12

� 603.2 in3 Simplify.

8 in.

13 in.

5 in.

h

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the volume of each cylinder. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

1 yd4 yd

10 cm

13 cm

20 ft

20 ft12 ft1.5 ft

18 cm2 cm2 ft

1 ft

Skills PracticeVolumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1


Less

on

13-

1

Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

Find the volume of each oblique prism or cylinder. Round to the nearest tenth ifnecessary.

7. 8.

5 in.

3 in.17 cm

18 cm

4 cm

6 yd

10 yd15 mm23 mm

16 in. 22 in.

34 in.

3 m

5 m

13 m

6 ft

8 ft

2 ft

18 cm

16 cm

8 cm


Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

AQUARIUM For Exercises 7–9, use the following information. Round answers tothe nearest tenth.Mr. Gutierrez purchased a cylindrical aquarium for his office. The aquarium has a height of 25�

12� inches and a radius of 21 inches.

7. What is the volume of the aquarium in cubic feet?

8. If there are 7.48 gallons in a cubic foot, how many gallons of water does the aquariumhold?

9. If a cubic foot of water weighs about 62.4 pounds, what is the weight of the water in theaquarium to the nearest five pounds?

30 cm

8 cm

13 yd

20 yd

10 yd

7 ft 25 ft16 mm 17.5 mm

5 in.

5 in.

5 in.

9 in.17 m

10 m

26 m

Practice Volumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1

Reading to Learn MathematicsVolumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1


Less

on

13-

1

Pre-Activity How is mathematics used in comics?

Read the introduction to Lesson 13-1 at the top of page 688 in your textbook.

In the cartoon, why was Shoe confused when the teacher said the class wasgoing to discuss volumes?

Reading the Lesson1. In each case, write a formula for the volume V of the solid in terms of the given variables.

a. a rectangular box with length a, width b, and height c

b. a rectangular box with square bases with side length x, and with height y

c. a cube with edges of length e

d. a triangular prism whose bases are isosceles right triangles with legs of length x, andwhose height is y

e. a prism whose bases are regular polygons with perimeter P and apothem a, andwhose height is h

f. a cylinder whose bases each have radius r, and whose height is three times the radiusof the bases

g. a regular octagonal prism in which each base has sides of length s and apothem a,and whose height is t

h. a cylinder with height h whose bases each have diameter d

i. an oblique cylinder whose bases have radius a and whose height is b

j. a regular hexagonal prism whose bases have side length s, and whose height is h

Helping You Remember2. A good way to remember a mathematical concept is to explain it to someone else. Suppose

that your younger sister, who is in eighth grade, is having trouble understanding whysquare units are used to measure area, but cubic units are needed to measure volume.How can you explain this to her in a way that will make it easy for her to understandand remember the correct units to use?


Visible Surface Area

Use paper, scissors, and tape to make five cubes that have one-inch edges.Arrange the cubes to form each shape shown. Then find the volume and the visible surface area. In other words, do not include the area of surfacecovered by other cubes or by the table or desk.

1. 2.

volume � volume �

surface area � surface area �

3. 4. 5.

volume � volume � volume �

surface area � surface area � surface area �

6. Find the volume and the visible surface area of the figure at the right.

volume �

surface area �

4 in.

4 in.

3 in.

3 in.

3 in.

8 in.

3 in.

5 in.

5 in.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1

Study Guide and InterventionVolumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2


Less

on

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2

Volumes of Pyramids This figure shows a prism and a pyramid that have the same base and the same height. It is clear that the volume of the pyramid is less than the volume of the prism. More specifically,the volume of the pyramid is one-third of the volume of the prism.

Volume of If a pyramid has a volume of V cubic units, a height of h units, a Pyramid and a base with an area of B square units, then V � �1

3�Bh.

Find the volume of the square pyramid.

V � �13�Bh Volume of a pyramid

� �13�(8)(8)10 B � (8)(8), h � 10

� 213.3 Multiply.

The volume is about 213.3 cubic feet.

Find the volume of each pyramid. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6. 6 yd

8 yd

5 yd15 in.

15 in.

16 in.

18 ft

regularhexagon 6 ft

4 cm8 cm

12 cm

10 ft

6 ft15 ft

12 ft

8 ft

10 ft

8 ft

8 ft

10 ft

ExercisesExercises

ExampleExample


Volumes of Cones For a cone, the volume is one-third the product of the height and the base. The base of a cone is a circle, so the area of the base is �r2.

Volume of a Right If a cone has a volume of V cubic units, a height of h units, Circular Cone and the area of the base is B square units, then V � �1

3�Bh.

The same formula can be used to find the volume of oblique cones.

Find the volume of the cone.

V � �13��r2h Volume of a cone

� �13��(5)212 r � 5, h � 12

� 314.2 Simplify.

The volume of the cone is about 314.2 cubic centimeters.

Find the volume of each cone. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

16 cm

45�26 ft

20 ft

45�18 yd

20 yd30 in.

12 in.

8 ft

10 ft6 cm10 cm

12 cm

5 cm

r

h


Volumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2

ExercisesExercises

ExampleExample

Skills PracticeVolumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2


Less

on

13-

2

Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

Find the volume of each oblique pyramid or cone. Round to the nearest tenth ifnecessary.

7. 8.

12 cm

6 cm

4 ft4 ft

6 ft

66�18 mm

25 yd

14 yd

25 m

12 m

8 in.10 in.

14 in.

4 cm7 cm

8 cm

5 ft5 ft

8 ft


Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

7. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is 4 metersin diameter, and the height of the shed is 3.8 meters. What is the volume of the shed?

8. HISTORY The start of the pyramid age began with King Zoser’s pyramid, erected in the27th century B.C. In its original state, it stood 62 meters high with a rectangular basethat measured 140 meters by 118 meters. Find the volume of the original pyramid.

37 ft11 ft

6 in.6 in.

11 in.

52�12 mm19 ft

9 ft

12.5 cm25 cm

23 cm

9.2 yd9.2 yd

13 yd

Practice Volumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2

Reading to Learn MathematicsVolumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2


Less

on

13-

2

Pre-Activity How do architects use geometry?


In addition to reflecting more light, why do you think the architect of theTransamerica Pyramid may have designed the building as a square pyramidrather than a rectangular prism?

Reading the Lesson1. In each case, two solids are described. Determine whether the first solid or the second

solid has the greater volume, or if the two solids have the same volume. (Answer bywriting first, second, or same.)a. First solid: A rectangular prism with length x, width y, and height z

Second solid: A rectangular prism with length 2x, width y, height zb. First solid: a rectangular prism that has a square base with side length x and that

has height ySecond solid: a square pyramid whose base has side length x and that has height y

c. First solid: a right cone whose base has radius x and that has height ySecond solid: an oblique cone whose base has radius x and that has height y

d. First solid: a cone whose base has radius x, and whose height is ySecond solid: a cylinder whose bases have radius x, and whose height is y

e. First solid: a cone whose base has radius x and whose height is ySecond solid: a square pyramid whose base has side length x and whose height is y

2. Supply the missing numbers to form true statements.

a. If the length, width, and height of a rectangular box are all doubled, its volume will

be multiplied by .

b. If the radius of a cylinder is tripled and the height is unchanged, the volume will be

multiplied by .

c. In a square pyramid, if the side length of the base is multiplied by 1.5 and the height

is doubled, the volume will be multiplied by .

d. In a cone, if the radius of the base is tripled and the height is doubled, the volume

will be multiplied by .

e. In a cube, if the edge length is multiplied by 5, the volume will be multiplied by .

Helping You Remember

3. Many students find it easier to remember mathematical formulas if they can put themin words. Use words to describe in one sentence how to find the volume of any pyramidor cylinder.


FrustumsA frustum is a figure formed when a plane intersects a pyramid orcone so that the plane is parallel to the solid’s base. The frustum is the part of the solid between the plane and the base. To find thevolume of a frustum, the areas of both bases must be calculated andused in the formula

V � �13

�h(B1 � B2 � �B1B2�),where h � height (perpendicular distance between the bases),B1 � area of top base, and B2 � area of bottom base.

Describe the shape of the bases of each frustum. Then find the volume. Round to the nearest tenth.

1. 2.

3. 4.

12 ft13 ft

7 ft8 m

6 m

12 m

4.5 m2.25 m

3 m

5 m

3 in.

7.5 in.

4.5 in.

13 cm

6 cm

9 cm

5 cm

19.5 cm

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2

Study Guide and InterventionVolumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3


Less

on

13-

3

Volumes of Spheres A sphere has one basic measurement, the length of its radius. If you know the radius of a sphere, you can calculate its volume.

Volume of a Sphere

If a sphere has a volume of V cubic units and a radius of r units, then V � �43

��r 3.

Find the volume of a sphere with radius 8 centimeters.

V � �43��r3 Volume of a sphere

� �43��(8)3 r � 8

� 2144.7 Simplify.

The volume is about 2144.7 cubic centimeters.

A sphere with radius 5 inches just fits inside a cylinder. What is the difference between the volume of thecylinder and the volume of the sphere? Round to the nearest cubic inch.The base of the cylinder is 25� in2 and the height is 10 in., so the volume of the cylinder is 250� in3. The volume of the sphere is �

43��(5)3

or �5030�� in3. The difference in the volumes is 250� � �

5030�� or about 262 in3.

Find the volume of each solid. Round to the nearest tenth.

1. 2. 3.

4. 5. 6.

7. A hemisphere with radius 16 centimeters just fits inside a rectangular prism. What isthe difference between the volume of the prism and the volume of the hemisphere?Round to the nearest cubic centimeter.

8 in. difference between volume of cube and volume of sphere

13 in.5 in.

8 cm

16 in.

6 in.

5 ft

5 in.

5 in.

5 in.5 in.

8 cm

r

ExercisesExercises

Example 1Example 1

Example 2Example 2


Solve Problems Involving Volumes of Spheres If you want to know if a spherecan be packed inside another container, or if you want to compare the capacity of a sphereand another shape, you can compare volumes.

Compare the volumes of the sphere and the cylinder. Determine which quantity is greater.

V � �43��r3 Volume of sphere V � �r2h Volume of cylinder

� �r2(1.5r) h � 1.5r

� 1.5�r3 Simplify.

Compare �43��r3 with 1.5�r3. Since �

43� is less than 1.5, it follows that

the volume of the sphere is less than the volume of the cylinder.

Compare the volume of a sphere with radius r to the volume of each figure below.Which figure has a greater volume?

1. 2.

3. 4.

5. 6.2a

r

3r

r

r3r

r

rr

rr2

2r

r1.5r


Volumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3

ExercisesExercises

ExampleExample

Skills PracticeVolumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3


Less

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Find the volume of each sphere or hemisphere. Round to the nearest tenth.

1. The radius of the sphere is 9 centimeters.

2. The diameter of the sphere is 10 inches.

3. The circumference of the sphere is 26 meters.

4. The radius of the hemisphere is 7 feet.

5. The diameter of the hemisphere is 12 kilometers.

6. The circumference of the hemisphere is 48 yards.

7. 8.

9. 10.

14.4 m

4.5 in.

94.8 ft16.2 cm


Find the volume of each sphere or hemisphere. Round to the nearest tenth.

1. The radius of the sphere is 12.4 centimeters.

2. The diameter of the sphere is 17 feet.

3. The circumference of the sphere is 38 meters.

4. The diameter of the hemisphere is 21 inches.

5. The circumference of the hemisphere is 18 millimeters.

6. 7.

8. 9.

10. PACKAGING Amber plans to ship a mini-basketball she bought for her nephew. Thecircumference of the ball is 24 inches and the package she wants to ship it in is arectangular box that measures 8 inches � 8 inches � 9 inches. Will the basketball fit inthe box? Explain.

C � 43 mm

32 m

C � 58 cm12.32 ft

Practice Volumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3

Reading to Learn MathematicsVolumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3


Less

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13-

3

Pre-Activity How can you find the volume of Earth?


How would you estimate the radius of Earth based on Eratosthenes’estimate of its diameter?

Reading the Lesson

1. Name all solids from the following list for which each volume formula can be used:prism, pyramid, cone, cylinder, sphere, hemisphere.

a. V � Bh b. V � �43��r3

c. V � �13�Bh d. V � �r2h

e. V � �13��r2h f. V � �

23��r3

2. Let r represent the radius and d represent the diameter of a sphere. Determine whethereach formula below can be used to find the volume of a sphere, a hemisphere, or neither.

a. V � �2�

3r3� b. V � �

16��d3

c. V � �13��r3 d. V � �

34��r3

e. V � ��1d2

3� f. V � �

43��r2h

3. Compare the volumes of these three solids. Then complete the sentence below.

Of the three solids shown above, the has the largest volume and the

has the smallest volume.


4. A good way to remember something is to explain it to someone else. Suppose that your classmate Loretta knows that the expressions �

43��r3 and 4�r2 are used in finding

measurements related to spheres, but can’t remember which one is used to find thesurface area of a sphere and which one is used to find the volume. How can you help herto remember which is which?

2r

r

rrr


Spheres and DensityThe density of a metal is a ratio of its mass to its volume. Forexample, the mass of aluminum is 2.7 grams per cubic centimeter.Here is a list of several metals and their densities.

Aluminum 2.7 g/cm3 Copper 8.96 g/cm3

Gold 19.32 g/cm3 Iron 7.874 g/cm3

Lead 11.35 g/cm3 Platinum 21.45 g/cm3

Silver 10.50 g/cm3

To calculate the mass of a piece of metal, multiply volume by density.

Find the mass of a silver ball that is 0.8 cm in diameter.

M � D V

� 10.5 �43��(0.4)3

� 10.5 (0.27)� 2.83

The mass is about 2.83 grams.

Find the mass of each metal ball described. Assume the balls are spherical. Round your answers to the nearest tenth.

1. a copper ball 1.2 cm in diameter

2. a gold ball 0.6 cm in diameter

3. an aluminum ball with radius 3 cm

4. a platinum ball with radius 0.7 cm

Solve. Assume the balls are spherical. Round your answers to the nearest tenth.

5. A lead ball weighs 326 g. Find the radius of the ball to the nearest tenth of a centimeter.

6. An iron ball weighs 804 g. Find the diameter of the ball to the nearest tenth of a centimeter.

7. A silver ball and a copper ball each have a diameter of 3.5 cm.Which weighs more? How much more?

8. An aluminum ball and a lead ball each have a radius of 1.2 cm.Which weighs more? How much more?

Enrichment

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ExampleExample

Study Guide and InterventionCongruent and Similar Solids

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Congruent or Similar Solids If the corresponding angles and sides of two solids arecongruent, then the solids are congruent. Also, the corresponding faces are congruent andtheir surface areas and volumes are equal. Solids that have the same shape but aredifferent sizes are similar. You can determine whether two solids are similar by comparingthe ratio, or scale factor, of corresponding linear measurements.

Describe each pair of solids.

• Figures I and II are similar because the figures have the same shape. The ratio of eachpair of corresponding sides is 1:3.

• Figures III and IV are congruent because they have the same shape and all correspondingmeasurements are the same.

• Figures V and VI are not congruent, and they are not similar because �48� �

1122�.

Determine whether each pair of solids are similar, congruent, or neither.

1. 2.

3. 4.

5. 6.2

7

21

6

5

8

5

8

4

4

88

5

5

2

2

2

26

6

7

7

12

4

5

1

106

8

53

4

I II III IV V VIsimilar congruent non-similar

12 5

5

5 12

12 4

85

5

5

7

7

9

6

4 32

ExercisesExercises

ExampleExample


Properties of Similar Solids These two solids are similar with a scale factor of 1:2. The surface areas are 62 cm2 and 248 cm2 and the volumes are 30 cm3 and 240 cm3. Notice that the ratio of the surface areas is 62:248, which is 1:4 or 12:22, and the ratio of the volumes is 30:240, which is 1:8 or 13:23.

If two solids are similar with a scale factor of a :b, then the surface areas have a ratio of a2:b2, and the volumes have a ratio of a3:b3.

Use the two spheres.a. Find the scale factor for the two spheres.

The scale factor for the two spheres is the same as the ratio of their radii, or 5:3.

b. Find the ratio of the surface areas of the two spheres.The ratio of the surface areas is 52:32 or 25:9.

c. Find the ratio of the volumes of the two spheres.The ratio of the volumes is 53:33 or 125:27.

Find the scale factor for each pair of similar figures. Then find the ratio of theirsurface areas and the ratio of their volumes.

1. 2.

3. 4.

5. 6.

8 6

5

3

1215

4 yd16 yd

15 m12 m

7 in. 4 in.

3 ft4 ft

5 cm3 cm

10 cm5 cm2 cm

3 cm

6 cm

4 cm


Congruent and Similar Solids

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ExercisesExercises

ExampleExample

Skills PracticeCongruent and Similar Solids

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1.

2.

3.

4.

For Exercises 5–8, refer to the two similar prisms.

5. Find the scale factor of the two prisms.

6. Find the ratio of the surface areas.

7. Find the ratio of the volumes.

8. Suppose the volume of the larger prism is 810 cubic centimeters. Find the volume of thesmaller prism.

15 cm12 cm

9 cm 10 cm8 cm

6 cm

18 in.

16 in.16 in.

9 in.

6 mm

6 mm

4 mm9 mm

12 ft12 ft

14 ft

20 ft20 ft

21 ft

20 cm20 cm

10 cm

4 cm40 cm

8 cm



1.

2.

3.

4.

For Exercises 5–8, refer to the two similar prisms.

5. Find the scale factor of the two prisms.

6. Find the ratio of the surface areas.

7. Find the ratio of the volumes.

8. Suppose the surface area of the larger prism is 2560 square meters. Find the surfacearea of the smaller prism.

9. MINIATURES Frank Lloyd Wright designed every aspect of the Imperial Hotel in Tokyo,including the chairs. The dimensions of a miniature Imperial Hotel chair are 6.25 inches �3 inches � 2.5 inches. If the scale of the replica is 1:6, what are the dimensions of theoriginal chair?

20 m

20 m

22 m

12 m

12 m

13.2 m

7.5 cm

20 cm

15 cm

4.5 cm

12 cm9 cm

18 ft24 ft

24 ft9 ft

12 m

12 m15 m

2.5 m

2 m9.6 m

25 in.

15 in.

30 in.20 in.

Practice Congruent and Similar Solids

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Reading to Learn MathematicsCongruent and Similar Solids

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Pre-Activity How are similar solids applied to miniature collectibles?


If you want to make a miniature with a scale factor of 1:64, how can youuse the actual object to find the measurements you should use to constructthe miniature?

Reading the Lesson1. Determine whether each statement is always, sometimes, or never true.

a. Two cubes are similar.b. Two cones are similar.c. Two cylinders in which the height is twice the diameter are similar.d. Two cylinders with the same volume are congruent.e. A prism with a square base and a square pyramid are similar.f. Two rectangular prisms with equal surface areas are similar.g. Nonsimilar solids have different volumes.h. Two hemispheres with the same radius are congruent.

2. Supply the missing ratios.

a. If the ratio of the diameters of two spheres is 3:1, then the ratio of their surface areas

is , and the ratio of their volumes is .

b. If the ratio of the radii of two hemispheres is 2:5, then the ratio of their surface areas

is , and the ratio of their volumes is .

c. If two cones are similar and the ratio of their heights is �43�, then the ratio of their

volumes is , and the ratio of their surface areas is .

d. If two cylinders are similar and the ratio of their surface areas is 100:49, then the

ratio of the radii of their bases is , and the ratio of their volumes is

.

Helping You Remember3. A good way to remember a new mathematical concept is to relate it to something you

already know. How can what you know about the units used to measure lengths, areas,and volumes help you to remember the theorem about the ratios of surface areas andvolumes of similar solids?


Congruent and Similar Solids

Determine whether each pair of solids is similar, congruent, or neither.

1. 2.

3. 4.

The two rectangular prisms shown at the right are similar.

5. Find the ratio of the perimeters of the bases.

6. What is the ratio of the surface areas?

7. Suppose the volume of the smaller prism is 60 in3.Find the volume of the larger prism.

Determine whether each statement is true or false. If the statement is false, rewrite it so that it is true.

8. If two cylinders are similar, then their volumes are equal.

9. Doubling the height of a cylinder doubles the volume.

10. Two solids are congruent if they have the same shape.

7 in.5 in.

24 yd

12 yd

12 yd6 yd

8 yd

16 yd12 m

3 m

3 m

3 m

3 m

3 m3 m

4 m

4 m4 m 4 m

4 m

10 m

48 m

16 m

15 m

14 cm

11 cm

7 cm

7 cm

Enrichment

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Study Guide and InterventionCoordinates in Space

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Graph Solids in Space In space, you can describe the location of a point using an ordered triple of realnumbers. The x-, y-, and z-axes are perpendicular to each other, and the coordinates for point P are the ordered triple (�4, 6, 5). A rectangular prism can bedrawn to show perspective.

Graph the rectangular solid that contains the ordered triple (2, 1, �2) and the origin. Label the coordinates of each vertex.• Plot the x-coordinate first. Draw a solid segment

from the origin 2 units in the positive direction.• Plot the y-coordinate next. Draw a solid segment

1 unit in the positive direction.• Plot the z-coordinate next. Draw a solid segment

2 units in the negative direction.• Draw the rectangular prism, using dotted lines for

hidden edges of the prism.• Label the coordinates of each vertex.

Graph the rectangular solid that contains the given point and the origin asvertices. Label the coordinates of each vertex.

1. A(2, 1, 3) 2. G(�1, 2, 3)

3. P(�2, 1, �1) 4. T(�1, 3, 2)

y

x

z

(0, 0, 0)

(0, 3, 0)

(�1, 3, 0)

(0, 0, 2) (0, 3, 2)

(�1, 0, 2)

(�1, 0, 0)

T (�1, 3, 2)

y

x

z

(0, 0, 0)

(0, 1, 0)

P(�2, 1, �1)

(�2, 0, �1) (�2, 1, 0)(�2, 0, 0)

(0, 0, �1) (0, 1, �1)

y

x

z

(0, 0, 0)

(0, 0, 3)

(�1, 2, 0)(�1, 0, 0)

(�1, 0, 3)G(�1, 2, 3)

(0, 2, 0)

(0, 2, 3)

y

x

z(0, 0, 3)

(0, 0, 0)

(2, 0, 3)

(2, 0, 0)(2, 1, 0)

(0, 1, 0)

(0, 1, 3)

A(2, 1, 3)

y

x

z

(0, 0, 0) (0, 1, 0)

(0, 1, �2)

(2, 1, �2)(2, 0, �2)

(0, 0, �2)

(2, 0, 0) (2, 1, 0)

y

x

z

O

P(�4, 6, 5)

ExercisesExercises

ExampleExample


Distance and Midpoint Formulas You can extend the Distance Formula and theMidpoint Formula to three dimensions to find the distance between two points in space and to find the midpoint of the segment connecting two points.

Distance Formula Given two points A(x1, y1, z1) and B(x2, y2, z2) in space, the distance between

in Space A and B is given by AB � �(x1 ��x2)2 �� (y1 �� y2)2 �� (z1 �� z2)2�.

Midpoint Formula Given two points A(x1, y1, z1) and B(x2, y2, z2) in space, the midpoint of A�B� is

in Space at ��x1 �

2x2�, �

y1 �

2y2�, �

z1 �

2z2��.

Determine the distance between A(3, 2, �5) and B(�4, 6, 9).Then determine the coordinates of the midpoint of A�B�.

AB � �(x1 ��x2)2 �� ( y1 �� y2)2 �� (z1 �� z2)2�� (3 � (��4))2�� (2 �� 6)2 �� (�5 �� 9)2�� 72 � (��4)2 �� (�14�)2�� 49 ��16 ��196�� 16.2

midpoint of A�B� � ��x1 �

2x2�, �

y1 �

2y2�, �

z1 �

2z2��

� ��3 �

2(�4)�, �

2 �2

6�, ��5

2� 9��

� (�0.5, 4, 2)

Determine the distance between each pair of points. Then determine thecoordinates of the midpoint M of the segment joining the pair of points.

1. A(0, 7, �4) and B(�2, 8, 3) 2. C(�7, 6, 5) and D(10, 2, �5)

3. E(3, 1, �2) and F(�2, 3, 4) 4. G(�4, 1, 1) and H(0, 2, �1)

5. J(6, 1, �2) and K(�1, �2, 1) 6. L(�5, 0, �3) and N(0, 0, �4)


Coordinates in Space

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ExercisesExercises

ExampleExample

Skills PracticeCoordinates in Space

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1. A(�5, 3, 2) 2. H(3, 2, 5)

3. Dilate the prism by a scale factor of 2. Graph the image under the dilation.


4. R(2, 1, 0) and S(3, 3, 4) 5. Q(5, 0, �2) and T(2, 3, 2)

6. A(�4, 1, 6) and B(�1, 0, 4) 7. J(0, 5, 1) and K(4, �3, 2)

A�B�

C� D�E�F�

G�H�

y

x

z

AB

C

G H

ED

F

N(0, 0, 0)

M(3, 0, 0)L(3, 2, 0)

P(0, 2, 0)

K(0, 2, 5)J(0, 0, 5)

I(3, 0, 5)

H(3, 2, 5)

A(�5, 3, 2)

E(�5, 3, 0)

B(�5, 0, 2)

C(0, 0, 2)D(0, 3, 2)

H(0, 3, 0)

G(0, 0, 0)

F (�5, 0, 0)



1. E(4, 6, �2) 2. R(�3, �5, 4)


3. Y(�5, 1, 2) and Z(3, �3, 1) 4. E(4, 2, 0) and F(3, 2, �2)

5. B(�2, �2, �3) and C(1, �3, 0) 6. H(2, 0, �3) and I(4, �1, 5)

7. ANIMATION Derek wants to animate an image for his science presentation by movingit from one position to another. The mesh of the image is a rectangular prism withcoordinates A(�3, 2, 3), B(�3, 0, 3), C(0, 0, 3), D(0, 2, 3), E(�3, 2, 0), F(�3, 0, 0), G(0, 0, 0),and H(0, 2, 0). Find the coordinates of the mesh after the translation (x, y, z) → (x � 7, y, z).Graph both the preimage and image of the mesh.

A�B�

C� D�

E�

F�

G� H�

AB

C D EFG

H

X(0, 0, 0)

Y(0, �5, 0)

V(�3, �5, 0)

R(�3, �5, 4) S(�3, 0, 4)

T(0, 0, 4)U(0, �5, 4)

W(�3, 0, 0)

K(0, 0, 0)

J(4, 0, 0)I(4, 6, 0)

L(0, 6, 0)

H(0, 6, �2)

E(4, 6, �2)F(4, 0, �2)

G(0, 0, �2)

Practice Coordinates in Space

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Reading to Learn MathematicsCoordinates in Space

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Pre-Activity How is three-dimensional graphing used in computer animation?


Why would a mesh be created first?

Reading the Lesson

1. Refer to the figure. Match each point from the first column with its coordinates from the second column.

a. A i. (3, 0, 0)

b. B ii. (3, 0, �4)

c. O iii. (3, �2, 0)

d. J iv. (3, �2, �4)

e. H v. (0, 0, 0)

f. K vi. (0, �2, 0)

g. T vii. (0, �2, �4)

h. R viii. (0, 0, �4)

2. Which of the following expressions give the distance between the points at (4, �1, �5)and (�3, 2, �9)?

A. �72 � (��3)2 �� 42� B. �12 � 1�2 � (��14)2�

C. �22 � 2�2 � 42� D. ��12�, �

12�, �7�

E. �(�3 �� 4)2 �� (�1 �� 2)2 �� (�9 �� 5)2� F. �24�

G. �(�3 �� 4)2 �� [2 ��(�1)]2� � [��9 � (��5)]2� H. �74�


3. A good way to remember new mathematical formulas is to relate them to ones youalready know. How can you use your knowledge of the Distance and Midpoint Formulasin two dimensions to remember the formulas in three dimensions?

y

x

z

A

K

BO

R

H

T

J


Planes and Cylindrical SurfacesConsider the points (x, y, z) in space whose coordinates satisfy the equation z � 1. Since x and y do not occur in the equation, any point with its z-coordinate equal to 1 has coordinates that satisfy the equation. These are the points in the plane 1 unit above the xy-plane. This plane is perpendicular to the z-axis at (0, 0, 1).

Next consider the points (x, y, z) whose coordinates satisfy x2 � y2 � 16. In the xy-plane,all points on the circle with center (0, 0, 0) andradius 4 have coordinates that satisfy the equation. In the plane perpendicular to the z-axis at (0, 0, k), the points that satisfy theequation are those on the circle with center (0, 0, k) and radius 4. The graph in space of x2 � y2 � 16 is an infinite cylindrical surface whose axis is the z-axis and whose radius is 4.

Describe the graph in space of each equation. You may find it helpful to make sketches on a separate sheet.

1. x � 5

2. y � �2

3. x � y � 7

4. z2 � y2 � 25

5. (x � 2)2 � (y � 5)2 � 1

6. x2 � y2 � z2 � 0

z

y

x

O

(0, 0, k)

plane for z � k

z

y

x

O

(0, 0, 1)N

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Chapter 13 Test, Form 11313


Ass

essm

entsWrite the letter for the correct answer in the blank at the right of each

question.

1. Which of the following could be the units of measure for the volume of a solid?A. cubic inches B. square inches C. inches D. cubic seconds

2. The area of the base of a prism is 96 square centimeters and the height is 9 centimeters. Find the volume.A. 288 cm3 B. 864 cm3 C. 932 cm3 D. 7776 cm3

3. The volume of a cylinder is 62.8 cubic meters and the radius is 2 meters. Findthe height to the nearest meter.A. 20 m B. 10 m C. 8 m D. 5 m

4. A cylinder has a radius that is 4 inches long and a height that is 9 inches long.Find the volume to the nearest tenth.A. 131.1 in3 B. 226.2 in3 C. 452.4 in3 D. 1809.6 in3

5. Find the volume of a pyramid with a height of 10 inches and a base with anarea of 21 square inches.A. 210 in3 B. 105 in3 C. 70 in3 D. 35 in3

6. Find the volume of the pyramid.A. 360 cm3 B. 390 cm3

C. 1080 cm3 D. 1170 cm3

7. A cone and a cylinder have the same radius and the same height. The volumeof the cone is what fraction of the volume of the cylinder?

A. �12� B. �

13� C. �

14� D. �

18�

8. Find the volume to the nearest tenth.A. 1206.4 in3 B. 402.1 in3

C. 301.6 in3 D. 100.5 in3

9. A sphere has a radius that is 12 centimeters long. Find the volume to thenearest tenth.A. 7238.2 cm3 B. 3619.1 cm3 C. 1809.6 cm3 D. 603.2 cm3

10. A sphere has a volume that is 36� cubic meters. Find the radius of the sphere.A. 2 m B. 3 m C. 6 m D. 12 m

11. The radius of a sphere is increased from 6 inches to 8 inches. How much isthe volume increased to the nearest tenth?A. 3719.6 in3 B. 1239.9 in3 C. 117.3 in3 D. 33.5 in3

6 in.

10 in.

10 cm9 cm

13 cm

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 13 Test, Form 1 (continued)1313

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. Which solid has the greater volume?A. sphere B. cylinderC. The volumes D. not enough

are equal. information

13. Which solid is similar to this solid?A. B.

C. D.

14. Which of the following describes the two spheres?A. congruent B. similarC. both A and B D. neither A nor B

15. The ratio of the side lengths of two cubes is 3:7. Find the ratio of their volumes.A. 3:7 B. 9:21 C. 9:49 D. 27:343

16. Two similar prisms have equilateral triangular bases and the edges of thebases are 50 centimeters and 20 centimeters. Find the ratio of the perimetersof the bases.A. 5�2�:2�5� B. 5:2 C. 25:4 D. 125:8

17. Find the coordinates of the image of A(�3, 5, 6) under the translation (x, y, z) → (x � 3, y � 5, z � 1).A. A�(0, 0, 5) B. A�(0, 10, 5) C. A�(�6, 0, �7) D. A�(�6, 10, 5)

18. The graph of the rectangular solid contains the origin and which other point?A. (3, 2, 4) B. (2, 4, 3)C. (4, 3, 2) D. (3, 4, 2)

For Questions 19 and 20, A(6, 5, 4) and B(�2, 4, 0).

19. Find the coordinates of the midpoint of A�B�.

A. �4, �12�, 2� B. (8, 1, 4) C. (4, 9, 4) D. �2, �

92�, 2�

20. Find the distance between A and B.A. �13� B. �17� C. 9 D. �113�

Bonus Describe the solid that could be formed by rotating this figure about line �. 3 cm

6 cm

�

y

x

z

9 ft 6 ft

15 in.

18 in.

15 in.15 in.

18 in.

12 in.

10 in.

10 in.18 in.

5 in.6 in.

3r

r

r

B:

NAME DATE PERIOD

Chapter 13 Test, Form 2A1313


Ass

essm


question.

1. How many cubic feet are in one cubic yard?A. 3 B. 9 C. 27 D. 81

2. The surface area of a cube is 96 square feet. Find the volume.A. 4 ft3 B. 16 ft3 C. 64 ft3 D. 256 ft3

3. A cylinder whose height is 5 meters has a volume of 320� cubic meters. Findthe radius of the cylinder.A. 8 m B. 12.8 m C. 64 m D. 201 m

4. A cylinder has a 10-inch diameter and an 11-inch height. Find the volume tothe nearest tenth.A. 172.8 in3 B. 345.6 in3 C. 863.9 in3 D. 3455.8 in3

5. A square pyramid has a height that is 8 centimeters long and a base withsides that are each 9 centimeters long. Find the volume.A. 648 cm3 B. 324 cm3 C. 216 cm3 D. 162 cm3

6. Find the volume to the nearest tenth.A. 80.0 ft3 B. 78.4 ft3

C. 48.0 ft3 D. 39.2 ft3

7. The volume of a cone is 1080� cubic centimeters and the radius is 18centimeters. Find the height.A. 5 cm B. 10 cm C. 20 cm D. 30 cm

8. Find the volume to the nearest tenth.A. 3619.1 m3 B. 4825.5 m3

C. 14,476.5 m3 D. 43,429.4 m3

9. A sphere has a 21-inch radius. Find the volume to the nearest tenth.A. 38,792.4 in3 B. 19,396.2 in3 C. 5541.8 in3 D. 1847.3 in3

10. A sphere has a volume that is 972� cubic inches. Find the radius.A. 2 in. B. 3 in. C. 6 in. D. 9 in.

11. A sphere has a 6-inch radius. A cone has a 12-inch height and base with a 6-inch radius. Compare their volumes.A. The volume of the sphere is greater.B. The volume of the cone is greater.C. The volumes are equal.D. not enough information

60�24 m

10 ft

10 ft6 ft

4 ft

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 13 Test, Form 2A (continued)1313

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. A golf ball has a 3.8-centimeter diameter and a tennis ball has a 7-centimeterdiameter. Find the difference between their volumes to the nearest tenth.A. 1206.9 cm3 B. 220.5 cm3 C. 150.9 cm3 D. 17.2 cm3

13. Two solids are congruent. Which statement about the two solids is true?A. They have a scale factor of 1.B. Their corresponding angles are congruent.C. The ratio of the corresponding linear measurements is 1:1.D. All of the above are true statements.

14. Two square pyramids are similar. The sides of the bases are 4 inches and 12 inches. The height of the smaller pyramid is 6 inches. Find the height ofthe larger pyramid.A. 24 in. B. 18 in. C. 16 in. D. 14 in.

15. The ratio of the radii of two similar cylinders is 3:5. The volume of the smallercylinder is 54� cubic centimeters. Find the volume of the larger cylinder.A. 90� cm3 B. 150� cm3 C. 250� cm3 D. 540� cm3

16. The ratio of the heights of two similar solids is 7:9. Find the ratio of theirvolumes.A. 7:9 B. 14:18 C. 49:81 D. 343:729


18. Find the coordinates of B(�9, 6, 12) under a dilation of factor �23�.

A. B��227�, 9, 18� B. B��

235�, �

230�, �

338��

C. B�(�3, 2, 4) D. B�(�6, 4, 8)

For Questions 19 and 20, C(4, 3, 2) and D(5, �1, �2).

19. Find the coordinates of the midpoint of C�D�.

A. ��12�, 2, 2� B. (�1, 4, 4) C. (9, 2, 0) D. ��

92�, 1, 0�

20. Find the distance between C and D.A. �85� B. �33� C. �11� D. �7�

Bonus A cone is formed by rolling up a 90° sector of a circle having an 8-inch radius. Find the volume of the cone to the nearest tenth.

8 in.8 in.

y

x

z

B:

NAME DATE PERIOD

Chapter 13 Test, Form 2B1313


Ass

essm


question.

1. Which of the following is the formula for the volume of a prism?A. V � 2�r2 � 2�rh B. V � phC. V � �r2 D. V � Bh

2. The lateral area of a cube is 324 square centimeters. Find the volume.A. 9 cm3 B. 81 cm3 C. 729 cm3 D. 972 cm3

3. Find the volume to the nearest tenth.A. 31.4 in3 B. 41.9 in3

C. 125.7 in3 D. 502.7 in3

4. A cylinder has a radius that is 7 inches long and a height that is 10 incheslong. Find the volume to the nearest tenth.A. 1539.4 in3 B. 490.0 in3 C. 219.9 in3 D. 70.0 in3

5. A right triangular pyramid has a 12-meter height and a base with legs thatare 3 meters and 4 meters long. Find the volume.A. 144 m3 B. 72 m3 C. 48 m3 D. 24 m3

6. The volume of a square pyramid is 100 cubic feet and the height is 10 feetlong. Find the length of a side of the base.A. 15 ft B. �30� ft C. 7.5 ft D. �5� ft

7. The volume of a cone is 336� cubic feet and the height is 7 feet long. Find theradius.A. 144 ft B. 36 ft C. 24 ft D. 12 ft

8. Find the volume to the nearest tenth.A. 41,224.0 m3 B. 20,612.0 m3

C. 10,306.0 m3 D. 763.4 m3

9. A sphere has a 48-centimeter diameter. Find the volume to the nearest tenth.A. 463,246.7 cm3 B. 57,905.8 cm3 C. 28,952.9 cm3 D. 7238.2 cm3

10. A sphere has a volume that is 288� cubic inches. Find the radius.A. 3 in. B. 6 in. C. 8 in. D. 12 in.

11. A sphere has a 10-centimeter radius and a cone has a 15-centimeter heightand a base with an 8-centimeter radius. Compare their volumes.A. The volume of the sphere is greater.B. The volume of the cone is greater.C. Their volumes are equal.D. not enough information

45�27 m

10 in.

4 in.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 13 Test, Form 2B (continued)1313

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. A beach ball has an 18-inch diameter and a kick ball has a 12-inch diameter.Find the difference of the volumes to the nearest tenth.A. 17,190.8 in3 B. 2148.8 in3 C. 188.5 in3 D. 113.1 in3

13. Two solids with the same shape but not necessarily the same size are .A. similar solidsB. congruent solidsC. congruent and similar solidsD. neither congruent nor similar solids

14. Find the scale factor between the two similar cones.A. 3:8 B. 1:3C. 1:2 D. 1:4

15. The ratio of the heights of two similar solids is 6:11. Find the ratio of theirsurface areas.A. 6:11 B. 36:121 C. 216:1331 D. 24:44

16. The ratio of the radii of two spheres is 7:3. The volume of the larger sphere is343� cubic feet. Find the volume of the smaller sphere.A. 147� ft3 B. 63� ft3 C. 36� ft3 D. 27� ft3


18. Find the image of C(�10, �6, 5) under the translation (x, y, z) → (x � 5, y � 8, z � 4).A. C�(�15, 2, 1) B. C�(�5, �14, 9)C. C�(�5, �14, �1) D. C�(50, �48, �20)

For Questions 19 and 20, E(8, �7, 6) and F(10, �8, 4).19. Find the distance between E and F.

A. �649� B. �13� C. 3 D. 1

20. Find the coordinates of the midpoint of EF.

A. �9, ��125�, 5� B. (18, �5, 10) C. ��1, �

12�, 1� D. (�2, 1, 2)

Bonus Find the volume of plastic needed to mold this tube to the nearest tenth.

2 cm

13 cm

1 cm

y

x

z

6 ft

16 ft8 ft

3 ft

?

B:

NAME DATE PERIOD

Chapter 13 Test, Form 2C1313


Ass

essm

ents1. A box has a length of 12 inches, a width of 9 inches, and a

height of 4 inches. Find the volume.

2. The volume of a rectangular prism is 120 cubic feet and thearea of the base is 60 square feet. Find the length of a lateraledge of the prism.

3. A cylinder has a 12-foot radius and a 17-foot height. Find thevolume to the nearest tenth.

4. Find the volume to the nearest tenth.

5. A regular hexagonal pyramid has a height that is 15 feet and a base 6 feet on each side. Find the volume.

6. The volume of a pyramid is 120 cubic inches and the area ofthe base is 50 square inches. Find the height.

7. The height of a cone is doubled and the radius is tripled. Howmany times greater is the volume?


9. A sphere has a diameter that is 7.36 inches long. Find thevolume to the nearest tenth.

10. The surface area of a sphere is 804.2 square inches. Find thevolume to the nearest cubic inch.

11. A sphere is circumscribed about an 8-inch by 8-inch by 8-inchcube. Find the volume of the sphere to the nearest cubic inch.


9 cm5 cm

30�

8 ft 5 ft

20 m

6 m

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

NAME DATE PERIOD

SCORE


Chapter 13 Test, Form 2C (continued)1313

13. Determine whether the statement Two cubes are similar isalways, sometimes, or never true.

14. Determine whether these two cylinders are congruent, similar,or neither.

15. The ratio of the heights of two similar prisms is 2:7. Thesurface area of the smaller prism is 50 square meters. Findthe surface area of the larger prism.

16. The ratio of the volumes of two similar solids is 8:27. Find theratio of their surface areas.

17. Find the image of D(11, �2, �9) under the translation (x, y, z) → (x � 4, y � 3, z � 6).

18. Graph the rectangular solid that contains the origin and thepoint A(1, 2, 3) as vertices.

For Questions 19 and 20, use a sphere that has a diameterwith endpoints X(�3, 4, 6) and Y(5, 6, 8).

19. Find the coordinates of the center of the sphere.

20. Find the radius of the sphere.

Bonus Find the volume of the solid formed by the rotation of�ABC about line m to the nearest tenth.

5 in.

AC

B13 in.

m

4 in.

10 in. 9 in.

3 in.

13.

14.

15.

16.

17.

18.

19.

20.

B:

y

x

z

A

NAME DATE PERIOD

Chapter 13 Test, Form 2D1313


Ass

essm

ents1. Find the volume.

2. An aquarium is 18 inches long, 8 inches wide, and 14 incheshigh. The water in it is 4 inches deep. Find the volume of the water.


4. A cylindrical can has a volume of 72� cubic centimeters and aheight that is 8 centimeters. Find the radius.

5. A square pyramid has a height that is 51 inches and a basewith sides that are each 11 inches long. Find the volume.

6. The volume of a pyramid is 216 cubic inches and the height is18 inches. Find the area of the base.

7. The height of a cone is tripled and the radius is doubled. Thevolume is how many times as great?

8. Find the volume of the solid to the nearest tenth.

9. A sphere has a radius that is 2.94 centimeters long. Find thevolume to the nearest tenth.

10. The surface area of a sphere is 1809.6 square feet. Find thevolume to the nearest cubic foot.

11. A box has a 6-inch length, a 6-inch width, and a 6-inch height.Find the volume of the largest ball that will fit in this box tothe nearest tenth.

4 cm

6 cm

25 cm7 cm

6 cm

5 cm

5 cm

12 cm

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 13 Test, Form 2D (continued)1313


13. Determine whether the statement Two congruent solids haveequal volumes is always, sometimes, or never true.

14. Determine whether these cubes are congruent, similar,or neither.

15. The ratio of the heights of two similar pyramids is 2:5 and thevolume of the smaller pyramid is 100 cubic feet. Find thevolume of the larger pyramid.

16. The ratio of the surface areas of two similar solids is 64:81.Find the ratio of their volumes.

17. Find the image of X(�7, 2, 5) under a dilation with scale factor 4.

18. Graph the rectangular solid that contains the origin and thepoint A(2, 4, 6) as vertices.

For Questions 19 and 20, use a sphere that has a diameterwith endpoints A(�5, 4, 10) and B(�11, 8, �2).

19. Find the length of a diameter of the sphere.

20. Find the coordinates of the center of the sphere.

Bonus Find the coordinates of the point that is three fourths ofthe way from A(�11, 9, 2) to B(4, 4, �8).

6 cm 8 cm

11 in. 1 in.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

y

x

z

A

NAME DATE PERIOD

Chapter 13 Test, Form 31313


Ass

essm

ents1. Find the volume of a cube with a 5-meter edge.

2. Find the volume.

3. Find the volume.

4. The volume of a cylinder is 96� cubic meters and the height is6 meters. Find the length of the diameter of this cylinder.

5. Sam is filling a rectangular pan with liquid from a cylindricalcan. The can is three-fourths full of water. Determine whetherall of the water will fit in the pan. Explain.

For Questions 6–10, find the volume to the nearest tenth.

6. The height of a regular octagonal pyramid is 12 centimetersand the apothem of the base is 4 centimeters.

7. 8.

9. 10.

11. A hemisphere has a radius that is 33 centimeters long. Findthe volume to the nearest tenth.

2 cm 8 cm

15 cm6 in.

3 in.

5 in.

2 in. 8 in.

13 in.

6 cm6 cm

6 cm

6 cm

8 in.6 in.

7 in.3 in.2 in.

2 in.

19 in.

3 cm2 cm

8 cm

10 cm

12 cm

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 13 Test, Form 3 (continued)1313

12. The radius of a sphere is multiplied by 4. How many timesgreater is the volume?

13. A cone is 9 centimeters deep and 4 centimeters across the top. Asingle scoop of ice cream, 4 centimeters in diameter, is placed ontop of the cone. If the ice cream melts into the cone, determinewhether the melted ice cream will fit in the cone. Explain.

14. Determine whether these two pyramids are congruent, similar, or neither.

15. The ratio of the volumes of two similar solids is 1:2. Find theratio of their surface areas.

16. The volumes of two spheres are in the ratio of 8:27. The radiusof the smaller sphere is 30 inches. Find the radius of thelarger sphere.

17. Dilate the cube by a factor of �12�. Then graph the image under

the dilation.

18. The center of a sphere is at A(5, �2, 8) and one endpoint of adiameter is at B(7, 5, 6). Find the coordinates of C, the otherendpoint of the diameter.

19. Find the distance between A(2�3�, �2, �2�) and B(3�3�, 1, 2�2�).

20. An airplane at an elevation of 1 mile is 20 miles east and 70 miles south of an airport. A second airplane at an elevation of

�12� mile is 50 miles west and 25 miles north of the airport. Find

the distance between the two airplanes to the nearest tenth.

Bonus A molding machine produces 4-centimeter cubes of plastic.The volume of molten plastic needed to produce these cubesis 11% more than the volume of plastic in the solid cubes.Find the amount of molten plastic needed to produce 100 cubes. (Note: 1 milliliter � 1 cubic centimeter.)

5 m

5 m

8 m

4 m

5 m

8 m6.4 m

8 m

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

y

x

z

NAME DATE PERIOD

Chapter 13 Open-Ended Assessment1313


Ass

essm

entsDemonstrate your knowledge by giving a clear, concise solution to

each problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.

1. Graph a rectangular prism and its image under a dilation with scale

factor �14�. Write the coordinates of the vertices of your original prism and

those of the image.

2. Write a formula for the volume of this solid in terms of the radius r.Explain how you determined your answer.

3. Give the dimensions of two cylinders in which the first has a greater volumethan the second, but the second has greater surface area than the first.

4. Draw and label the dimensions of a prism and a pyramid that have thesame volume.

5. Demonstrate how the formula for the volume of a cylinder can be obtainedfrom the formula for the volume of a prism.

6. Find one set of coordinates for A and C so that their midpoint is B(3, 4, 5).

45�

2r

r

r

NAME DATE PERIOD

SCORE


Chapter 13 Vocabulary Test/Review1313

Determine whether each sentence is true or false. If false,replace the underlined word or formula to make a truesentence.

1. Congruent solids have the same shape but not the same size.

2. Similar solids have the same shape and the same size.

3. An ordered pair gives the coordinates of a point in space.

4. Volume is the measure of the amount of space that a figureencloses.

5. V � �r2h is the formula for the volume of a cone.

6. V � 4�r2 is the formula for the volume of a sphere.

7. V � Bh is the formula for the volume of a prism.

8. V � �13�Bh is the formula for the volume of a pyramid.

9. V � ��r

3

2h� is the formula for the volume of a cylinder.

10. The formula for the distance between two points in space is

d � �(x1 ��x2)2 �� ( y1 �� y2)2 �� (z1 �� z2)2�.

In your own words—

11. Explain how to find the ratio of the volume of two similar solidsif a:b is the ratio of the lengths of the corresponding edges.

12. Explain how to find the midpoint of (x1, y1, z1) and (x2, y2, z2).

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

congruent solids ordered triple similar solids volume

NAME DATE PERIOD

SCORE

Chapter 13 Quiz (Lessons 13–1 and 13–2)

1313


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

1. A rectangular prism has a length of 16 feet, a width of 9 feet,and height of 8 feet. Find the volume.

2. The volume of a rectangular prism is 198 cubic centimeters,the length is 11 centimeters, and the height is 9 centimeters.Find the width.

3. A cylinder has a diameter of 20 inches and height of 9 inches.Find the volume to the nearest tenth.

4. A pyramid has a height of 18 centimeters and a base with anarea of 26 square centimeters. Find the volume.

5. Find the volume to the nearest tenth. 16 cm

17 cm

Chapter 13 Quiz (Lesson 13–3)

1313

1.

2.

3.

4.

5.

1. The radius of a sphere is doubled. How many times greater isits volume?

2. A hemisphere has a base with an area that is 25� squarecentimeters. Find the volume to the nearest tenth.

3. A sphere is inscribed in a cube whose volume is 1728 cubicmillimeters. Find the volume of the sphere to the nearest tenth.


5. STANDARDIZED TEST PRACTICE A sphere has a radiusthat is 15.6 inches long. Find the volume to the nearest tenth.A. 1019.4 in3 B. 7951.2 in3

C. 15,902.4 in3 D. 47,707.2 in3

8.1 mm2.3 mm

NAME DATE PERIOD

SCORE



1313

1.

2.

3.

4.

5.

1. Determine whether the cylinders are congruent, similar, or neither.

2. Determine whether solids having a scale factor of 1 arecongruent, similar, or neither.

3. The ratio of the heights of two similar cones is 2:3. Find theratio of their surface areas.

4. The scale factor of a model house is 1:100. The doors in thehouse are 84 inches high. How tall are the doors in the model?

5. Determine whether the statement A cone and a cylinder maybe similar is true or false.

10 m

12 m

3 m

4 m

NAME DATE PERIOD

SCORE


1313

1.

2.

3.

4.

5.

1. Find the coordinates of a pyramid with vertices A(2, 4, 0),B(3, 5, 0), C(�1, 6, 0), and D(2, 5, 3) under the translation (x, y, z) → (x � 3, y � 7, z � 5).

2. Graph the rectangular solid that contains the origin and B(5, 1, 3) as vertices.

For Questions 3–5, A(�2, 4, 9) and B(6, 1, 10).

3. Find the coordinates of the midpoint of A�B�.

4. Find the distance between A and B.

5. If B is the midpoint of A�C�, find the coordinates of C.

y

x

z

B

NAME DATE PERIOD

SCORE

Chapter 13 Mid-Chapter Test (Lessons 13–1 through 13–3)

1313


Ass

essm

ents

1. Choose the formula for the volume of a cylinder.

A. V � �r2h B. V � ��r

3

2h� C. V � 4�r2 D. V � �

4�3r3�

2. The volume of a cube is 216 cubic meters. Find the length of an edge.A. 72 m B. 36 m C. 12 m D. 6 m

3. A rectangular aquarium is 18 inches long and 12 inches wide and contains1620 cubic inches of water. Find the depth of the water.A. 15 in. B. 10 in.C. 7.5 in. D. 3.75 in.

4. How does the volume of a cone change when its radius is doubled?A. The volume is multiplied by 2. B. The volume is multiplied by 4.C. The volume is multiplied by 8. D. The volume is divided by 2.

5. The surface area of a sphere is equal to its volume. Find the radius.A. 1 unit B. 3 unitsC. 9 units D. cannot be determined

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part II

6. Find the volume of this bead to the nearest centimeter.

7. Find the volume of a square pyramid whose slant height is 13 centimeters and whose base has sides that are each 24 centimeters long.


9. A sphere has a surface area that is 169� square centimeters.Find the volume to the nearest tenth.

10. Find the volume.

4 in.

4 in.

4 in.

8 in.

42�

2 in.

2 cm2 cm

2 cm

1 cm

Part I Write the letter for the correct answer in the blank at the right of each question.


Chapter 13 Cumulative Review(Chapters 1–13)

1313

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

1. If �C and �D are congruent supplementary angles and m�C � 3x � 72, find x. (Lesson 2-8)

2. Determine the slope of the line that contains J(24, �5) andK(16, �18). (Lesson 3-3)

3. Classify �HJK by its sides if its vertices are H(6, �2),J(�9, �10), and K(�9, 6). (Lesson 4-1)

4. Find the measures of one exterior angle and one interior angleof a convex regular pentagon. (Lesson 8-1)

5. A 10-centimeter by 6-centimeter rectangle is enlarged by afactor of 3. What are the dimensions of the new rectangle?(Lesson 9-5)

6. In �A, mBC�� 2x � 16 and m�BAC � 5x � 98. Find x.

(Lesson 10-2)

7. Write an equation of a circle with center at (1, �2) and a diameter of 8 units. (Lesson 10-8)

8. Find the area of a rhombus with diagonals that are 24 centimeters and 78 centimeters long. (Lesson 11-2)

9. Write the number of faces and the shapes of the faces of a hexahedron. (Lesson 12-1)

10. Find the lateral area of a regular pentagonal prism if theheight of the prism is 9 centimeters and the perimeter of thepentagonal base is 35 centimeters. (Lesson 12-3)

11. Find the volume to the nearest tenth.(Lesson 13-1)

12. Find the volume to the nearest tenth.(Lesson 13-2)

12 in.7 in.

6 in.

3 m

12 m

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–13)


1. If D�G� bisects �EDF, which is a true statement? (Lesson 5-1)

A. B is the incenter of �DEF.B. DB � BGC. B�E� � B�F�D. B is equidistant from D�E� and D�F�.

2. If S�T� is a midsegment of �PQR,which is a true statement? (Lesson 6-4)

E. PQ � 2ST F. 2PQ � STG. 3PQ � 4ST H. 4PQ � 3ST

3. Which matrix can be used to find the coordinates of the image of a triangle reflected in the origin? (Lesson 9-7)

A. B. C. D.

4. Find the circumference of �G. (Lesson 10-1)

E. 9� in. F. 12� in.G. 15� in. H. 30� in.

5. If a line is tangent to a circle, then it is to the radiusdrawn to the point of tangency. (Lesson 10-5)

A. perpendicular B. parallelC. congruent D. not related

6. Find the area of a circle with a diameter of 28 meters. (Lesson 11-3)

E. 49� m2 F. 196� m2

G. 784� m2 H. 3136� m2

7. Which figures are needed to make a net for the solid? (Lesson 12-2)

A. 2 trapezoids and 4 rectanglesB. 6 rectanglesC. 4 trapezoids and 2 rectanglesD. 6 trapezoids

8. Find the volume of a right circular cone with a height of 14 inchesand a diameter of 10 inches to the nearest tenth. (Lesson 13-2)

E. 183.3 in3 F. 366.5 in3 G. 733.0 in3 H. 1466.1 in.

?

G9 in.

12 in.

1 00 1

�1 0 0 �1

�1 0 0 1

1 00 �1

Q

T

SP

R

E

GB

D

F

NAME DATE PERIOD

SCORE 1313

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

1.

2.

3.

4.

5.

6.

7.

8. E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

Ass

essm

ents


Standardized Test Practice (continued)

9. Find m�HJK. (Lesson 10-6)

10. Find the y-coordinate of the image of A(3, �5)under a rotation of 180° clockwise about theorigin. (Lesson 9-3)

11. Find x so that the figures are similar. (Lesson 13-4)

8 cmx cm

9 cm

12 cm

JH

K276�

NAME DATE PERIOD

1313

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

Part 3: Short Response

Instructions: Show your work or explain in words how you found your answer.

9. 10.

11.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

12. Quadrilateral PQRS is inscribed in �O so that �S and �Qare opposite angles, m�Q � 10x � 11, and m�S � 6x � 27.Find x. (Lesson 10-4)

13. Find the area of quadrilateral CDEFif the distance from C to D�F� is 14meters. (Lesson 11-2)

14. A right cylinder has a height of 7 inches and a base with adiameter of 8 inches. What is the lateral area to the nearesttenth? (Lesson 12-4)

15. Find the volume of a hemisphere with a diameter of 18 feet tothe nearest tenth. (Lesson 13-3)

16. Determine the coordinates of the midpoint of the segmentjoining W(2, 4, �9) and X(�1, 7, �3). (Lesson 13-5)

7 m

12 m

D E

F

C

12.

13.

14.

15.

16.

4 2 5

3

Unit 4 Review (Chapters 11–13)

1313


Ass

essm

ents1. Determine whether BCDE with vertices B(3, 5), C(6, �2),

D(�4, �2), and E(�7, 5) is a square, rectangle, orparallelogram. Then find the area of BCDE.

2. If the trapezoid has an area of 152 square centimeters, find h.

3. A regular hexagon is inscribed in �P.Find the exact area of the shadedregion.

4. Find the area of the irregular figure.

5. Find the probability that a point selected at random lies in the shaded region.

For Questions 6 and 7, refer to the figure.

6. Identify the solid and name its bases.

7. Find the surface area of the solid to thenearest tenth if AB � 8, AC � 25,CF � 14, and EF � 25.

8. How many hexagons would be drawn in the net for ahexagonal prism?

9. Find the lateral area of a rectangular prism with base 4.8 inches by 6.2 inches and height 5.9 inches.

10. The surface area of a right cylinder is 324.7� square meters.Find the radius of the base of the cylinder if the height of thecylinder is 10.6 meters.

B

E F

C

A

D

x

y

O

(�1, 7) (2, 7)

(6, 2)(2, 1)(�2, 1)

P

18 in.

P

12 cm

20 cm

h

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Unit 4 Review (continued)1313

11. Find the surface area of the regular pentagonal pyramid to the nearest tenth.

12. Mark is cutting colored paper to make conical party hats for his daughter’s birthday party. He wants to make them with a diameter of 9 inches and a height of 12 inches. What is the lateral area of one party hat?

13. To the nearest tenth, find the outer surface area of a bowl inthe shape of a perfect hemisphere with diameter 36 centimeters.

14. Sandra is packing a box with dimensions 54 inches by 78 inches by 42 inches. What is the maximum volume in cubicfeet that this box can hold? Round to the nearest tenth.

15. Find the volume of a cone with radius 11 inches and height 15 inches to the nearest tenth.

16. Which figure has the greater volume, the sphere or thecylinder?

17. Determine whether the pair of cones is similar, congruent,or neither.

18. Graph the rectangular solid that contains M(�1, �4, �3) andthe origin as vertices. Label the coordinates of each vertex.

24 cm7.5 cm

22.5 cm

36 cm

6 in.

2 in.

10 in.

9 in.

12 in.

4 ft

10 ft

11.

12.

13.

14.

15.

16.

17.

18.

y

x

z

(0, 0, 0)

(0, 0, �3)

(�1, �4, 0)(�1, 0, 0)

(�1, 0, �3)

M(�1, �4, �3)

(0, �4, 0)

(0, �4, �3)

NAME DATE PERIOD

Second Semester Test(Chapters 8–13)

1313


Ass

essm

entsFor Questions 1–17, write the letter for the correct answer in the blank

at the right of each question.

1. What are the coordinates of the intersection of the diagonals of parallelogramDEFG with vertices D(�1, �4), E(�2, 4), F(2, 6), and G(3, �2)?A. (0.5, 1) B. (0, 0)C. (0, 5) D. (2.5, 2)

2. If JKML is a rectangle, JN � 2a � 12, and LN � 5a � 48, find JM.A. 12 B. 24C. 52 D. 104

3. Which of the following statements is always true?A. A parallelogram is a rectangle. B. A square is a parallelogram.C. A rectangle is a square. D. A rhombus is a square.

4. If ABDC is a trapezoid, find AB so that E�F�is the median of ABDC.A. 13 B. 20C. 30.5 D. 61

5. Triangle LMN with vertices L(2, �5), M(�1, 0), N(0, 7) is reflected in the x-axis. Find the coordinates of the vertices of the reflected image.A. L�(2, �5), M�(�1, 0), N�(0, 7) B. L�(�2, �5), M�(1, 0), N�(0, 7)C. L�(2, 5), M�(�1, 0), N�(0, �7) D. L�(�5, 2), M�(0, �1), N�(7, 0)

6. Trapezoid WXYZ is rotated with respect to lines � and m . Determine the angle of rotation.A. 160° clockwiseB. 80° counterclockwiseC. 40° counterclockwiseD. 25° clockwise

7. Triangle HJK has vertices H(15, �9), J(�3, 0), and K(12, 6). Find thecoordinates of the vertices for a dilation centered at the origin with a scale

factor of ��13�.

A. H�(�45, 27), J�(9, 0), K�(�36, �18)B. H�(�5, 3), J�(1, 0), K�(�4, �2)C. H�(�9, 15), J�(0, �3), K�(6, 12)D. H�(15, 9), J�(�3, 0), K�(12, �6)

8. Find the exact circumference of �H.A. 42� B. 56�

C. 70� D. 84�H

42

56

W�

X�Y�

Z�

W

m �

XY

Z

A B

E F

C D34

27

J KN

L M

1.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE


Second Semester Test (continued)1313

9.

10.

11.

12.

13.

14.

15.

16.

17.

For Questions 9 and 10, refer to the figure.The diameter of �A is 40.

9. Find the exact value of BC if AC � 12.A. 16 B. 18C. 20 D. 24

10. If DF � 26�23�, find the exact value of EF so that D�F� is tangent to �A.

A. 8 B. 12 C. 13�13� D. 14�

23�

11. Which circle has equation (x � 5)2 � y2 � 81?A. �A with center at (5, 0), radius 5 B. �B with center at (0, 5), radius 9C. �C with center at (5, 0), radius 9 D. �D with center at (0, 5), radius 5

12. A parallelogram has a base of 21 inches, sides 14 inches, and height 9 inches.Find the area and perimeter of the parallelogram.A. area � 189 in2, perimeter � 70 in. B. area � 126 in2, perimeter � 70 in.C. area � 126 in2, perimeter � 49 in. D. area � 189 in2, perimeter � 79 in.

13. Find the area of a regular hexagon with a perimeter of 75 centimeters to thenearest tenth.A. 11.3 cm2 B. 12.5 cm2 C. 405.9 cm2 D. 450 cm2

14. Which of the following is a base of the figure?A. ONM B. ONJLC. OMKL D. KMNJ

15. Which of the following is the surface area to the nearest tenth of a rightcylinder with a height of 24 feet and a base diameter of 18 feet?A. 2714.3 ft2 B. 4750.1 ft2

C. 1357.2 ft2 D. 1866.1 ft2

16. Find the volume of the sphere to the nearest tenth.A. 289.5 m2 B. 463.2 m2

C. 1158.1 m2 D. 3705 m2

17. The solids are similar with a scale factor of 4:5.Find the ratio of the volumes of the twopyramids.A. 7:8 B. 12:15C. 16:25 D. 64:125

11.259

12 15

4.8 m

J

L

O

M

KN

A

B

CD

E F

NAME DATE PERIOD



Ass

essm

ents18. Find the sum of the measures of the interior angles and the

measure of one exterior angle of a convex regular 12-gon.

19. Find x and y so that ABCD is a parallelogram.

20. Determine whether JKLM with vertices J(0, �5), K(�4, �2),L(�1, 2), and M(3, �1) is a rhombus, a rectangle, or a square.List all that apply. Explain your reasoning.

21. Name the missing coordinates for the isosceles trapezoid.

22. Graph quadrilateral DEFG with vertices D(1, �4), E(3, �1),F(4, �2), and G(4, �4) and its image under the translation (x, y) → (x � 4, y � 3).

23. Find the measure of the preimage X�Y� if X�Y� � 204 in thedilated image and the scale factor is 8.

24. A charter boat travels a path due south at 20 miles per hour.The current is flowing from the east at 4 miles per hour. Whatis the resultant speed and direction of the boat? Round to thenearest tenth.

For Questions 25 and 26, refer to the figure. In �M, G�J� ⊥ M�L�.

25. Find the measure of each numbered angle in the circle.

26. Suppose a segment were drawn from K to G. What would the measure of �GKH be?

G

HJ

K

L

M22�1

2 3

4

x

y

(b, a)

(–c, 0)

(0, a)

(?, ?)

B C

A 4x � 30

2y � 1 3y � 5

x 6

D

18.

19.

20.

21.

22.

23.

24.

25.

26.

D�

E�F�

G�

D

EF

G

x

y

O

NAME DATE PERIOD

SCORE




27. If mVT�� 74 and mQU�

� 23, findm�P.

28. Solve for RS if VS � 32, ST � 25, and QS � 16.

29. Find the area of rhombus QRST with vertices Q(�2, 4),R(3, 1), S(�2, �2), and T(�7 , 1).


30. Find the area of the figure to thenearest tenth.

31. What is the probability that a point selected at random lies on the shaded square?

32. Find the surface area of the prism.

33. A regular pyramid has a lateral area of 468 squarecentimeters and a slant height of 9 centimeters. Find theperimeter of the base of this pyramid.

34. Which has the greater surface area, a right cone with a baseradius of 6 meters and a height of 6 meters or a sphere with aradius of 6 meters?

35. A rectangular box has dimensions 45 inches by 74 inches by38 inches, and a cylindrical barrel has base diameter 50 inchesand height of 68 inches. Which container holds more volume?

36. Find the volume of the pyramid.

37. Determine the exact distance between A(�2, 5, �8) and Z(5, 7, 12).

5 cm

6 cm

12 cm

18

249

12 mm

4.5 mm

PU

Q

RT

V

S

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

NAME DATE PERIOD

Final Test1313


Ass

essm

entsFor Questions 1–26, write the letter for the correct answer in the blank

at the right of each question.

1. What are the coordinates of the midpoint of T�D� with T(�1, 5) and D(3, �9)?A. (2, 7) B. (2, �2) C. (1, 7) D. (1, �2)

2. Given A(2, 5), B(�7, 5), and C(102, 5), which statement has the same truthvalue as the statement A, B, and C are collinear?A. AC � 100 B. �ABC is an obtuse triangle.C. The perimeter of �ABC is 109 units. D. A�B� � B�C�

3. Gary knows that if two angles of a triangle are congruent, then the triangle isisosceles. He is given �FGH with �G � �H. With just this information, Garycan use the Law of Detachment to conclude which of the following?A. �F is the vertex angle of �FGH. B. �FGH is an isosceles triangle.C. m�G � m�H D. G�F� � H�F�

4. Which angles are consecutive interior angles?A. �1 and �4 B. �2 and �8C. �3 and �6 D. �3 and �5

5. Which equation represents a line that contains R(�5, 4) and S(7, 2)?A. 5y � 3x � 19 B. y � 6x � 40 C. 6y � �x � 19 D. 5y � �4x

6. Triangle RST is isosceles with vertex angle S. RS is six times a number, RT istwo times the number, and ST is twelve more than three times the number.Find the perimeter of �RST.A. 72 units B. 56 units C. 24 units D. 4 units

7. In the figure, W and V are the midpoints of X�Y�and Y�Z�, respectively, �YXV � �YZW, and XY � ZY. Which postulate or theorem can you use to prove �WXU � �VZU?A. AAS B. SSSC. SAS D. Congruence cannot be determined.

8. Given 3y � 8 � 14, which statement would you assume to write an indirectproof to show that y � 2?A. y � 2 B. 3y � 8 � 14 C. y � 2 D. 3y � 8 � 14

9. In �QRS, QR � 42 and RS � 50. Which of the following cannot be SQ?A. 9 B. 75 C. 84 D. 92

10. There are 135 girls in a club of 415 students. Find the ratio of girls to boys inthe club.A. 27:56 B. 13:41 C. 1:3 D. 3:17

Y

W X

ZVU

B1 2

3 45

68

7

m

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Final Test (continued)1313

11.

12.

13.

14.

15.

16.

17.

18.

11. Mr. Trout’s class visited New York City with a 6-foot-tall scale model thatthey made of the Chrysler building. The class measured the model’s shadowat 1.44 inches and then measured the shadow of the Chrysler building at250.8 inches. What is the height of the actual building in feet?A. 125.4 ft B. 418 ft C. 836 ft D. 1045 ft

12. Slippery Ski Slope has an angle of depression of 9.5 degrees and is 412 meters long. Find thevertical drop of the slope to the nearest meter.A. 68 m B. 69 mC. 402 m D. 406 m

13. Find x to the nearest tenth.A. 49.2 B. 56.6C. 86.3 D. 98.4

14. If PQSR is a rhombus, and m�QRP � 73,find m�QSP.A. 73 B. 68C. 34 D. 17

15. What are the coordinates of B if BCDE is an isosceles trapezoid with verticesC(�3, 5), D(�1, 7), and E(1, 7)?A. B(1, 5) B. B(3, 5) C. B(3, �5) D. B(�1, 6)

16. Determine the direction of CD� to the nearest degree given C(�3, 10) and D(5, �12).A. 110° B. 290° C. 23 units D. 548 units

17. Triangle DEF has vertices D(4, 5), E(2, 7) and F(�3, 8). Which of the followingdemonstrates scalar multiplication to dilate �DEF so that its perimeter isdouble its original perimeter?

A. B. 2

C. 2 D.

18. A circle has a radius of 26 millimeters. What is the circumference of the circleto the nearest tenth?A. 81.7 mm B. 163.4 mm C. 530.9 mm D. 2123.7 mm

4 2 �35 7 8

1 00 1

4 2 �35 7 8

4 5 2 7�3 8

4 2 �35 7 8

0 22 0

P Q

SR

49�75

x

9.5�

? 412 m

NAME DATE PERIOD



Ass

essm

ents19. Segments P�R� and Q�R� are tangent to �M.

Which of the following is true?A. �QPR � �PQR B. P�Q� � P�R�C. m�R � 75 D. PQ � QR

20. A group of 1800 single women in their twenties were asked to choose which type of date they would prefer from three choices. The results of the survey are recorded in the circle graph.What is the measure of the central anglerepresenting the “Rock Climbing” category?A. 28.8 B. 86.4C. 99.5 D. 111.6

21. Trapezoid JKLM has an area of 84 square centimeters. Find the height of JKLM.A. 2.2 cm B. 3.6 cmC. 7 cm D. 9 cm

22. Find the area of the parallelogram to the nearest tenth.A. 43.9 ft2 B. 56.2 ft2

C. 79.2 ft2 D. 112 ft2

23. What is the lateral area of the cylinder?A. 529� in2 B. 828� in2

C. 1886� in2 D. 9522� in2

24. Find the approximate amount of material needed to cover the entire solid.A. 371.6 m2 B. 678.6 m2

C. 923.6 m2 D. 1304.2 m2

25. A block of wood is cut with the dimensions shown. What is the volume of the wood?A. 2072 cm3 B. 2424 cm3

C. 3520 cm3 D. 7040 cm3

26. Two similar pyramids have slant heights of 15 and 24. What is the ratio ofthe surface areas of the two pyramids?A. 5:8 B. 25:64 C. 31:49 D. 125:512

20 cm

11 cm

32 cm

7 m

14 m

46 in.

18 in.

14 ft

8 ft45�

J K

LM

9 cm

15 cm

8%No response

ThemePark24%

RockClimbing

31%

DinnerCruise37%

Preferred Date

RM

Q

P 19.

20.

21.

22.

23.

24.

25.

26.

NAME DATE PERIOD

SCORE




27. Name a point in the interior of �KJG.

28. If m�GMK � 4x � 6 and m�KMJ � 3x � 32, find x and m�KMJ.

29. Identify the hypothesis and conclusion of the statement A circle with a radius of r has a circumference of 2�r.

30. Determine whether the following statement is always,sometimes, or never true.If two angles of a triangle measure 34 and 35, then the thirdangle must measure 111.

31. Graph the line that contains K(3, 4) and is perpendicular toGH�� containing G(�4, 3) and H(�1, 2).

32. Find x so that p || q.

33. If �RST � �BCD, find y and z.

For Questions 34–38, complete the proof.

Given: A�G� � E�H�C�A� � C�E��CGH � �CHG

Prove: �AGC � �EHCProof:Statements Reasons1. 1.2. G�C� � H�C� 2.3. 3.

39. When the medians of a triangle intersect, what is the point ofintersection called?

(Question 38)(Question 37)

(Question 36)

(Question 35)(Question 34)

A E

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C

y 9

z � 1

17

31

22

R T

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HK

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

K(3, 4)

NAME DATE PERIOD



Ass

essm

ents40. Write an inequality to describe the

possible values of y.

For Questions 41 and 42, refer to the figure. U�V� is a midsegment of�RST, and W�T� bisects �RTS.

41. If m�SRT � 7a and m�RUV � 29a,find a and m�VUT.

42. If RT � 8, UT � 4 and WT � 14,find XT.

43. Find a, b, and c.

44. In the figure, LK � 52. Find HKand LH.

45. Find the measure of each angle of the quadrilateral.

46. Find x and y so that quadrilateral LMNP is a parallelogram.

47. Determine how many lines of symmetry a regular pentagonhas. Then tell whether a regular pentagon has rotationalsymmetry.

48. Find the translation that moves triangle 1 to triangle 2 and then the translation that moves triangle 2 to triangle 3.

x

y

O

(�1, 4)

(2, �3)

(3, 0)

1

2

3

3x�

5y�39�

55�

P L

MN

(5x � 8)� (x 34)�

(3x 45)� (9x � 35)�G F

H E

135�KH

L

20

16 b

a

c

R TU

S

W V

X

98� 84�

10

88

y 740.

41.

42.

43.

44.

45.

46.

47.

48.

NAME DATE PERIOD

SCORE



49. In �B, WV� � YT� and YZ � 28.What is the length of D�X�?

50. Find m�1 and m�2.

51. To the nearest tenth, find the area of a regular pentagon witha perimeter of 142 feet.

52. The diameter of the circle is 40 centimeters.Find the area of the circle. Then find theprobability that a point selected at randomlies within a shaded region of the circle.

53. Draw the corner view of the figure given the orthogonaldrawing.

54. Find the surface area of the regular pyramid to the nearest tenth.

55. Compare the volumes of the hemisphere and semicylinder. Which solid has the greater volume?

56. Determine the distance between H(�2, 4, �9) and J(3, �5, �12). Then state the coordinates of the midpoint of H�J�.

9 ft

9 ft 10 ft

12 cm

16 cm

top view left view front view right view

78�56�

43�

31�93�

59�

G46�

178�

1 2

B

V X

Z

Y

T

C

DW

49.

50.

51.

52.

53.

54.

55.

56.

NAME DATE PERIOD

Standardized Test PracticeStudent Record Sheet (Use with pages 724–725 of the Student Edition.)

1313

© Glencoe/McGraw-Hill A1 Glencoe Geometry

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6 DCBADCBA

DCBADCBADCBA

DCBADCBADCBA

NAME DATE PERIOD

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Open-EndedPart 3 Open-Ended

Solve the problem and write your answer in the blank.

For Questions 13 and 14, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.

9 13 14

10

11

12

13 (grid in)

14 (grid in)

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

Record your answers for Questions 15–16 on the back of this paper.


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Answers (Lesson 13-1)


An

swer

s

Skil

ls P

ract

ice

Volu

mes

of

Pri

sms

and

Cyl

ind

ers

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ME

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____

____

____

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13-1

13-1

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Lesson 13-1

Fin

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he

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me

of e

ach

pri

sm o

r cy

lin

der

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nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

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cm

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80 in

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d t

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me

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sm o

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sary

.

1.2.

2040

m3

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3

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yd

360

31.9

cm

3

AQ

UA

RIU

MF

or E

xerc

ises

7–9

,use

th

e fo

llow

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info

rmat

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.Rou

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an

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the

nea

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th.

Mr.

Gu

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ical

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13-1

13-1



Readin

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13-1

13-1

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Lesson 13-1

Pre-

Act

ivit

yH

ow i

s m

ath

emat

ics

use

d i

n c

omic

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 13

-1 a

t th

e to

p of

pag

e 68

8 in

you

r te

xtbo

ok.

In t

he

cart

oon

,wh

y w

as S

hoe

con

fuse

d w

hen

th

e te

ach

er s

aid

the

clas

s w

asgo

ing

to d

iscu

ss v

olu

mes

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amp

le a

nsw

er:T

he

teac

her

was

refe

rrin

g t

o t

he

mat

hem

atic

al m

ean

ing

of

volu

me,

wh

ich

is t

he

amo

un

t o

f sp

ace

encl

ose

d b

y a

thre

e-d

imen

sio

nal

fig

ure

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oe

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th

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ab

ou

t vo

lum

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s b

oo

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hic

h is

a c

om

ple

tely

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Less

on

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eac

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rite

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orm

ula

for

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volu

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way

th

at w

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Vis

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show

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fin

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En

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NA

ME

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IOD

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13-1

13-1



An

swer

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Stu

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nte

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Volu

mes

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ME

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13-2

13-2

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Lesson 13-2

Vo

lum

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ram

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Th

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igu

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pris

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ram

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e sa

me

base

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d th

e sa

me

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ght.

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s cl

ear

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th

e vo

lum

e of

th

e py

ram

id i

s le

ss t

han

th

e vo

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e of

th

e pr

ism

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e sp

ecif

ical

ly,

the

volu

me

of t

he

pyra

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one-

thir

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th

e vo

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th

e pr

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.

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me

of

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a v

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vol

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ME

____

____

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13-2

13-2

Exer

cises

Exer

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Exam

ple

Exam

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Skil

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ract

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Volu

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ME

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____

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____

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13-2

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Lesson 13-2

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m

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m

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Fin

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wit

h a

rec

tan

gula

r ba

seth

at m

easu

red

140

met

ers

by 1

18 m

eter

s.F

ind

the

volu

me

of t

he

orig

inal

pyr

amid

.

abo

ut

341,

413.

3 m

3

37 ft

11 ft

6 in

.6

in.

11 in

.

52�

12 m

m19

ft

9 ft

12.5

cm

25 c

m

23 c

m

9.2

yd9.

2 yd

13 y

d

Pra

ctic

e (

Ave

rag

e)

Volu

mes

of

Pyr

amid

s an

d C

on

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-2

13-2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csVo

lum

es o

f P

yram

ids

and

Co

nes

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-2

13-2

©G

lenc

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cGra

w-H

ill73

3G

lenc

oe G

eom

etry

Lesson 13-2

Pre-

Act

ivit

yH

ow d

o ar

chit

ects

use

geo

met

ry?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 13

-2 a

t th

e to

p of

pag

e 69

6 in

you

r te

xtbo

ok.

In a

ddit

ion

to

refl

ecti

ng

mor

e li

ght,

wh

y do

you

th

ink

the

arch

itec

t of

th

eT

rans

amer

ica

Pyr

amid

may

hav

e de

sign

ed t

he b

uild

ing

as a

squ

are

pyra

mid

rath

er t

han

a r

ecta

ngu

lar

pris

m?

Sam

ple

an

swer

:Th

e p

yram

id is

mo

re u

nu

sual

an

d h

as a

mo

re d

ram

atic

ap

pea

ran

ce,s

o it

attr

acts

mo

re a

tten

tio

n.W

ith

th

e sh

arp

po

int

at t

he

top

,it

seem

s to

so

ar u

p in

to t

he

sky.

Rea

din

g t

he

Less

on

1.In

eac

h c

ase,

two

soli

ds a

re d

escr

ibed

.Det

erm

ine

wh

eth

er t

he

firs

t so

lid

or t

he

seco

nd

soli

d h

as t

he

grea

ter

volu

me,

or i

f th

e tw

o so

lids

hav

e th

e sa

me

volu

me.

(An

swer

by

wri

tin

g fi

rst,

seco

nd

,or

sam

e.)

a.F

irst

sol

id:A

rec

tan

gula

r pr

ism

wit

h l

engt

h x

,wid

th y

,an

d h

eigh

t z

Sec

ond

soli

d:A

rec

tan

gula

r pr

ism

wit

h l

engt

h 2

x,w

idth

y,h

eigh

t z

seco

nd

b.

Fir

st s

olid

:a r

ecta

ngu

lar

pris

m t

hat

has

a s

quar

e ba

se w

ith

sid

e le

ngt

h x

and

that

has

hei

ght

yS

econ

d so

lid:a

squ

are

pyra

mid

who

se b

ase

has

side

leng

th x

and

that

has

hei

ght

yfir

stc.

Fir

st s

olid

:a r

igh

t co

ne

wh

ose

base

has

rad

ius

xan

d th

at h

as h

eigh

t y

Sec

ond

soli

d:an

obl

iqu

e co

ne

wh

ose

base

has

rad

ius

xan

d th

at h

as h

eigh

t y

sam

ed

.F

irst

sol

id:a

con

e w

hos

e ba

se h

as r

adiu

s x,

and

wh

ose

hei

ght

is y

Sec

ond

soli

d:a

cyli

nde

r w

hos

e ba

ses

hav

e ra

diu

s x,

and

wh

ose

hei

ght

is y

seco

nd

e.F

irst

sol

id:a

con

e w

hos

e ba

se h

as r

adiu

s x

and

wh

ose

hei

ght

is y

Sec

ond

solid

:a s

quar

e py

ram

id w

hose

bas

e ha

s si

de le

ngth

xan

d w

hose

hei

ght

is y

first

2.S

upp

ly t

he

mis

sin

g n

um

bers

to

form

tru

e st

atem

ents

.

a.If

th

e le

ngt

h,w

idth

,an

d h

eigh

t of

a r

ecta

ngu

lar

box

are

all

dou

bled

,its

vol

um

e w

ill

be m

ult

ipli

ed b

y .

b.

If t

he

radi

us

of a

cyl

inde

r is

tri

pled

an

d th

e h

eigh

t is

un

chan

ged,

the

volu

me

wil

l be

mu

ltip

lied

by

.

c.In

a s

quar

e py

ram

id,i

f th

e si

de l

engt

h o

f th

e ba

se i

s m

ult

ipli

ed b

y 1.

5 an

d th

e h

eigh

t

is d

oubl

ed,t

he

volu

me

wil

l be

mu

ltip

lied

by

.

d.

In a

con

e,if

th

e ra

diu

s of

th

e ba

se i

s tr

iple

d an

d th

e h

eigh

t is

dou

bled

,th

e vo

lum

e

wil

l be

mu

ltip

lied

by

.

e.In

a c

ube

,if

the

edge

len

gth

is

mu

ltip

lied

by

5,th

e vo

lum

e w

ill

be m

ult

ipli

ed b

y .

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stu

den

ts f

ind

it e

asie

r to

rem

embe

r m

ath

emat

ical

for

mu

las

if t

hey

can

pu

t th

emin

wor

ds.U

se w

ords

to

desc

ribe

in

on

e se

nte

nce

how

to

fin

d th

e vo

lum

e of

an

y py

ram

idor

cyl

inde

r.S

amp

le a

nsw

er:

Mu

ltip

ly t

he

area

of

the

bas

e by

th

e h

eig

ht

and

div

ide

by 3

.

125

18

4.5

9

8

©G

lenc

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ill73

4G

lenc

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eom

etry

Fru

stu

ms

A f

rust

um

is a

fig

ure

for

med

wh

en a

pla

ne

inte

rsec

ts a

pyr

amid

or

con

e so

th

at t

he

plan

e is

par

alle

l to

th

e so

lid’

s ba

se.T

he

fru

stu

m i

s th

e pa

rt o

f th

e so

lid

betw

een

th

e pl

ane

and

the

base

.To

fin

d th

evo

lum

e of

a f

rust

um

,th

e ar

eas

of b

oth

bas

es m

ust

be

calc

ula

ted

and

use

d in

th

e fo

rmu

la

V�

�1 3� h(B

1�

B2

��

B1B

2�

),w

her

e h

�h

eigh

t (p

erpe

ndi

cula

r di

stan

ce b

etw

een

th

e ba

ses)

,B

1�

area

of

top

base

,an

d B

2�

area

of

bott

om b

ase.

Des

crib

e th

e sh

ape

of t

he

bas

es o

f ea

ch f

rust

um

.Th

en f

ind

th

e vo

lum

e.R

oun

d t

o th

e n

eare

st t

enth

.

1.2.

rect

ang

les;

617.

5 cm

3

circ

les;

335.

8 in

3

3.4.

trap

ezo

ids;

151.

6 m

3

circ

les;

3480

.9 f

t3

12 ft

13 ft

7 ft

8 m

6 m

12 m

4.5

m2.

25 m3

m

5 m

3 in

.

7.5

in.

4.5

in.

13 c

m 6 cm

9 cm

5 cm

19.5

cm

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-2

13-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Volu

mes

of

Sp

her

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-3

13-3

©G

lenc

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cGra

w-H

ill73

5G

lenc

oe G

eom

etry

Lesson 13-3

Vo

lum

es o

f Sp

her

esA

sph

ere

has

on

e ba

sic

mea

sure

men

t,th

e le

ngt

h o

f it

s ra

diu

s.If

you

kn

ow t

he

radi

us

of a

sph

ere,

you

can

cal

cula

te

its

volu

me.

Volu

me

of

a S

ph

ere

If a

sphe

re h

as a

vol

ume

of V

cubi

c un

its a

nd a

rad

ius

of r

units

, th

en V

��4 3� �

r3.

Fin

d t

he

volu

me

of a

sp

her

e w

ith

rad

ius

8 ce

nti

met

ers.

V�

�4 3� �r3

Vol

ume

of a

sph

ere

��4 3� �

(8)3

r�

8

�21

44.7

Sim

plify

.

Th

e vo

lum

e is

abo

ut

2144

.7 c

ubi

c ce

nti

met

ers.

A s

ph

ere

wit

h r

adiu

s 5

inch

es j

ust

fit

s in

sid

e a

cyli

nd

er.W

hat

is

the

dif

fere

nce

bet

wee

n t

he

volu

me

of t

he

cyli

nd

er a

nd

th

e vo

lum

e of

th

e sp

her

e? R

oun

d t

o th

e n

eare

st

cub

ic i

nch

.T

he

base

of

the

cyli

nde

r is

25�

in2

and

the

hei

ght

is 1

0 in

.,so

th

e vo

lum

e of

th

e cy

lin

der

is 2

50�

in3 .

Th

e vo

lum

e of

th

e sp

her

e is

�4 3� �(5

)3

or �50

30� �in

3 .T

he

diff

eren

ce i

n t

he

volu

mes

is

250�

��50

30� �or

abo

ut

262

in3 .

Fin

d t

he

volu

me

of e

ach

sol

id.R

oun

d t

o th

e n

eare

st t

enth

.

1.2.

3.

523.

6 ft

345

2.4

in3

8578

.6 in

3

4.5.

6.

268.

1 cm

357

6.0

in3

243.

9 in

3

7.A

hem

isph

ere

wit

h r

adiu

s 16

cen

tim

eter

s ju

st f

its

insi

de a

rec

tan

gula

r pr

ism

.Wh

at i

sth

e di

ffer

ence

bet

wee

n t

he

volu

me

of t

he

pris

m a

nd

the

volu

me

of t

he

hem

isph

ere?

Rou

nd

to t

he

nea

rest

cu

bic

cen

tim

eter

.78

05 c

m3

8 in

.di

ffere

nce

betw

een

volu

me

of c

ube

and

volu

me

of s

pher

e

13 in

.5

in.

8 cm

16 in

.

6 in

.

5 ft

5 in

.

5 in

.

5 in

.5

in.

8 cmr

Exer

cises

Exer

cises

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

©G

lenc

oe/M

cGra

w-H

ill73

6G

lenc

oe G

eom

etry

Solv

e Pr

ob

lem

s In

volv

ing

Vo

lum

es o

f Sp

her

esIf

you

wan

t to

kn

ow i

f a

sph

ere

can

be

pack

ed i

nsi

de a

not

her

con

tain

er,o

r if

you

wan

t to

com

pare

th

e ca

paci

ty o

f a

sph

ere

and

anot

her

sh

ape,

you

can

com

pare

vol

um

es.

Com

par

e th

e vo

lum

es o

f th

e sp

her

e an

d

the

cyli

nd

er.D

eter

min

e w

hic

h q

uan

tity

is

grea

ter.

V�

�4 3� �r3

Vol

ume

of s

pher

eV

��

r2h

Vol

ume

of c

ylin

der

��

r2(1

.5r)

h�

1.5r

�1.

5�r3

Sim

plify

.

Com

pare

�4 3� �r3

wit

h 1

.5�

r3.S

ince

�4 3�is

les

s th

an 1

.5,i

t fo

llow

s th

at

the

volu

me

of t

he

sph

ere

is l

ess

than

th

e vo

lum

e of

th

e cy

lin

der.

Com

par

e th

e vo

lum

e of

a s

ph

ere

wit

h r

adiu

s r

to t

he

volu

me

of e

ach

fig

ure

bel

ow.

Wh

ich

fig

ure

has

a g

reat

er v

olu

me?

1.2.

Th

e vo

lum

e o

f th

e h

emis

ph

ere

Th

e vo

lum

e o

f th

e sp

her

e is

gre

ater

.is

gre

ater

.

3.4.

Th

e vo

lum

e o

f th

e sp

her

eT

he

volu

me

of

the

sph

ere

is

gre

ater

.is

gre

ater

.

5.6.

Th

e vo

lum

e o

f th

e cy

lind

erca

nn

ot

be

det

erm

ined

(If

a�

0.63

,is

gre

ater

.th

e vo

lum

e o

f th

e h

emis

ph

ere

isg

reat

er.I

f a

�0.

63,t

he

volu

me

of

the

sph

ere

is g

reat

er.)

2ar

3r

r

r3r

r

rr

rr 2

2r

r1.

5r

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Volu

mes

of

Sp

her

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-3

13-3

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Volu

mes

of

Sp

her

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-3

13-3

©G

lenc

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w-H

ill73

7G

lenc

oe G

eom

etry

Lesson 13-3

Fin

d t

he

volu

me

of e

ach

sp

her

e or

hem

isp

her

e.R

oun

d t

o th

e n

eare

st t

enth

.

1.T

he

radi

us

of t

he

sph

ere

is 9

cen

tim

eter

s.

3053

.6 c

m3

2.T

he

diam

eter

of

the

sph

ere

is 1

0 in

ches

.

523.

6 in

3

3.T

he

circ

um

fere

nce

of

the

sph

ere

is 2

6 m

eter

s.

296.

8 m

3

4.T

he

radi

us

of t

he

hem

isph

ere

is 7

fee

t.

718.

4 ft

3

5.T

he

diam

eter

of

the

hem

isph

ere

is 1

2 ki

lom

eter

s.

452.

4 km

3

6.T

he

circ

um

fere

nce

of

the

hem

isph

ere

is 4

8 ya

rds.

933.

8 yd

3

7.8.

2226

.1 c

m3

446,

091.

2 ft

3

9.10

.

190.

9 in

378

1.7

m3

14.4

m

4.5

in.

94.8

ft16

.2 c

m

©G

lenc

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ill73

8G

lenc

oe G

eom

etry

Fin

d t

he

volu

me

of e

ach

sp

her

e or

hem

isp

her

e.R

oun

d t

o th

e n

eare

st t

enth

.

1.T

he

radi

us

of t

he

sph

ere

is 1

2.4

cen

tim

eter

s.

7986

.4 c

m3

2.T

he

diam

eter

of

the

sph

ere

is 1

7 fe

et.

2572

.4 f

t3

3.T

he

circ

um

fere

nce

of

the

sph

ere

is 3

8 m

eter

s.

926.

6 m

3

4.T

he

diam

eter

of

the

hem

isph

ere

is 2

1 in

ches

.

2424

.5 in

3

5.T

he

circ

um

fere

nce

of

the

hem

isph

ere

is 1

8 m

illi

met

ers.

49.2

mm

3

6.7.

7832

.9 f

t332

94.8

cm

3

8.9.

8578

.6 m

367

1.3

mm

3

10.P

AC

KA

GIN

GA

mbe

r pl

ans

to s

hip

a m

ini-

bask

etba

ll s

he

bou

ght

for

her

nep

hew

.Th

eci

rcu

mfe

ren

ce o

f th

e ba

ll i

s 24

in

ches

an

d th

e pa

ckag

e sh

e w

ants

to

ship

it

in i

s a

rect

angu

lar

box

that

mea

sure

s 8

inch

es �

8 in

ches

�9

inch

es.W

ill

the

bask

etba

ll f

it i

nth

e bo

x? E

xpla

in.

Yes;

the

dia

met

er o

f th

e b

all i

s ab

ou

t 7.

64 in

.,so

th

e b

all w

ill f

it in

th

e b

ox.

C �

43

mm

32 m

C �

58

cm12

.32

ft

Pra

ctic

e (

Ave

rag

e)

Volu

mes

of

Sp

her

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-3

13-3



Readin

g t

o L

earn

Math

em

ati

csVo

lum

es o

f S

ph

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ME

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13-3

13-3

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Lesson 13-3

Pre-

Act

ivit

yH

ow c

an y

ou f

ind

th

e vo

lum

e of

Ear

th?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 13

-3 a

t th

e to

p of

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e 70

2 in

you

r te

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How

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ld y

ou e

stim

ate

the

radi

us

of E

arth

bas

ed o

n E

rato

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enes

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tim

ate

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ts d

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eter

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rom

th

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ula

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wh

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ula

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an b

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plet

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nte

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bel

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thre

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lids

sh

own

abo

ve,t

he

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th

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rges

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lum

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d th

e

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emb

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goo

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ay t

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kn

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sion

s �4 3� �

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indi

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mem

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e is

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d to

fin

d th

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wh

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f r

in t

he

two

exp

ress

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he

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ress

ion

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h r

3w

ill g

ive

a m

easu

rem

ent

incu

bic

un

its,

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is t

he

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ress

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fo

r vo

lum

e.T

he

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ress

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2

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tio

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ts m

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to i

ts v

olu

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ple,

the

mas

s of

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um

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r cu

bic

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tim

eter

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ere

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lis

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thei

r de

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ver

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he

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iply

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um

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ram

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l d

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um

e th

e b

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n i

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s 80

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f th

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ll t

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sil

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n a

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and

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ME

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13-3

13-3

Exam

ple

Exam

ple



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Co

ng

ruen

t an

d S

imila

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s

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ME

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13-4

13-4

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Lesson 13-4

Co

ng

ruen

t o

r Si

mila

r So

lids

If t

he

corr

espo

ndi

ng

angl

es a

nd

side

s of

tw

o so

lids

are

con

gru

ent,

then

th

e so

lids

are

con

gru

ent.

Als

o,th

e co

rres

pon

din

g fa

ces

are

con

gru

ent

and

thei

r su

rfac

e ar

eas

and

volu

mes

are

equ

al.S

olid

s th

at h

ave

the

sam

e sh

ape

but

are

diff

eren

t si

zes

are

sim

ilar

.You

can

det

erm

ine

wh

eth

er t

wo

soli

ds a

re s

imil

ar b

y co

mpa

rin

gth

e ra

tio,

or s

cale

fac

tor,

of c

orre

spon

din

g li

nea

r m

easu

rem

ents

.

Des

crib

e ea

ch p

air

of s

olid

s.

•F

igu

res

I an

d II

are

sim

ilar

bec

ause

th

e fi

gure

s h

ave

the

sam

e sh

ape.

Th

e ra

tio

of e

ach

pair

of

corr

espo

ndi

ng

side

s is

1:3

.•

Fig

ure

s II

I an

d IV

are

con

gru

ent

beca

use

th

ey h

ave

the

sam

e sh

ape

and

all

corr

espo

ndi

ng

mea

sure

men

ts a

re t

he

sam

e.•

Fig

ure

s V

an

d V

I ar

e n

ot c

ongr

uen

t,an

d th

ey a

re n

ot s

imil

ar b

ecau

se �4 8�

�1 12 2�

.

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ine

wh

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er e

ach

pai

r of

sol

ids

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ila

r,co

ng

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t,or

nei

ther

.

1.2.

sim

ilar

nei

ther

3.4.

con

gru

ent

con

gru

ent

5.6.

nei

ther

sim

ilar

2 7

21

6

5

8

5

8

4

4

88

5

5

2

2

2 26

6

7

7

12

4

5

1

106

8

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4

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VV

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mila

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ent

non-

sim

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125

5

512

124

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5

5

7

7

9

6

43

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Exam

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Pro

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Sim

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Solid

sT

hes

e tw

o so

lids

ar

e si

mil

ar w

ith

a s

cale

fac

tor

of 1

:2.T

he

surf

ace

area

s ar

e 62

cm

2an

d 24

8 cm

2an

d th

e vo

lum

es a

re

30 c

m3

and

240

cm3 .

Not

ice

that

th

e ra

tio

of t

he

surf

ace

area

s is

62

:248

,wh

ich

is

1:4

or 1

2:2

2 ,an

d th

e ra

tio

of t

he

volu

mes

is

30:2

40,w

hic

h i

s 1:

8 or

13:2

3 .

If tw

o so

lids

are

sim

ilar

with

a s

cale

fac

tor

of a

:b,

then

the

sur

face

ar

eas

have

a r

atio

of

a2:b

2 , a

nd t

he v

olum

es h

ave

a ra

tio o

f a

3:b

3 .

Use

th

e tw

o sp

her

es.

a.F

ind

th

e sc

ale

fact

or f

or t

he

two

sph

eres

.T

he

scal

e fa

ctor

for

th

e tw

o sp

her

es i

s th

e sa

me

as

the

rati

o of

th

eir

radi

i,or

5:3

.

b.

Fin

d t

he

rati

o of

th

e su

rfac

e ar

eas

of t

he

two

sph

eres

.T

he

rati

o of

th

e su

rfac

e ar

eas

is 5

2:3

2or

25:

9.

c.F

ind

th

e ra

tio

of t

he

volu

mes

of

the

two

sph

eres

.T

he

rati

o of

th

e vo

lum

es i

s 53

:33

or 1

25:2

7.

Fin

d t

he

scal

e fa

ctor

for

eac

h p

air

of s

imil

ar f

igu

res.

Th

en f

ind

th

e ra

tio

of t

hei

rsu

rfac

e ar

eas

and

th

e ra

tio

of t

hei

r vo

lum

es.

1.2.

3:4;

9:16

;27

:64

7:4;

49:1

6;34

3:64

3.4.

4:5;

16:2

5;64

:125

2:1;

4:1;

8:1

5.6.

5:4;

25:1

6;12

5:64

1:2:

1:4;

1:8

86

5

3

1215

4 yd

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15 m

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7 in

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in.

3 ft

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6 cm

4 cm

Stu

dy G

uid

e a

nd I

nte

rven

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(con

tinued

)

Co

ng

ruen

t an

d S

imila

r S

olid

s

NA

ME

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____

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ER

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13-4

13-4

Exer

cises

Exer

cises

Exam

ple

Exam

ple



Skil

ls P

ract

ice

Co

ng

ruen

t an

d S

imila

r S

olid

s

NA

ME

____

____

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____

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13-4

13-4

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Lesson 13-4

Det

erm

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wh

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er e

ach

pai

r of

sol

ids

are

sim

ila

r,co

ng

ruen

t,or

nei

ther

.

1.si

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r

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mila

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ng

ruen

t

For

Exe

rcis

es 5

–8,r

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the

two

sim

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pri

sms.

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ind

the

scal

e fa

ctor

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the

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ind

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e su

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e ar

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the

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.

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.

6 m

m

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Det

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1.co

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r

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mila

r

For

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es 5

–8,r

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to

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e fa

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the

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e su

rfac

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ind

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e vo

lum

es.

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upp

ose

the

surf

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area

of

the

larg

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2560

squ

are

met

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d th

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spec

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th

e Im

peri

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in

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dim

ensi

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min

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mpe

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are

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ches

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inch

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f th

e sc

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he

repl

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is 1

:6,w

hat

are

th

e di

men

sion

s of

th

eor

igin

al c

hai

r?

37.5

in.�

18 in

.�15

in.

20 m

20 m

22 m

12 m

12 m

13.2

m

7.5

cm

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m

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m

4.5

cm

12 c

m9

cm

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24 ft

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12 m

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m

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m

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e (

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Co

ng

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t an

d S

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s

NA

ME

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____

____

____

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AT

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ER

IOD

____

_

13-4

13-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csC

on

gru

ent

and

Sim

ilar

So

lids

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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ER

IOD

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13-4

13-4

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eom

etry

Lesson 13-4

Pre-

Act

ivit

yH

ow a

re s

imil

ar s

olid

s ap

pli

ed t

o m

inia

ture

col

lect

ible

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 13

-4 a

t th

e to

p of

pag

e 70

7 in

you

r te

xtbo

ok.

If y

ou w

ant

to m

ake

a m

inia

ture

wit

h a

sca

le f

acto

r of

1:6

4,h

ow c

an y

ouu

se t

he

actu

al o

bjec

t to

fin

d th

e m

easu

rem

ents

you

sh

ould

use

to

con

stru

ctth

e m

inia

ture

?S

amp

le a

nsw

er:T

ake

linea

r m

easu

rem

ents

of

the

actu

al o

bje

ct.D

ivid

e ea

ch m

easu

rem

ent

by 6

4 to

fin

d t

he

corr

esp

on

din

g m

easu

rem

ent

for

the

min

iatu

re.

Rea

din

g t

he

Less

on

1.D

eter

min

e w

het

her

eac

h s

tate

men

t is

alw

ays,

som

etim

es,o

r n

ever

tru

e.a.

Tw

o cu

bes

are

sim

ilar

.al

way

sb

.T

wo

con

es a

re s

imil

ar.

som

etim

esc.

Tw

o cy

lin

ders

in

wh

ich

th

e h

eigh

t is

tw

ice

the

diam

eter

are

sim

ilar

.al

way

sd

.T

wo

cyli

nde

rs w

ith

th

e sa

me

volu

me

are

con

gru

ent.

som

etim

ese.

A p

rism

wit

h a

squ

are

base

an

d a

squ

are

pyra

mid

are

sim

ilar

.n

ever

f.T

wo

rect

angu

lar

pris

ms

wit

h e

qual

su

rfac

e ar

eas

are

sim

ilar

.so

met

imes

g.N

onsi

mil

ar s

olid

s h

ave

diff

eren

t vo

lum

es.

som

etim

esh

.T

wo

hem

isph

eres

wit

h t

he

sam

e ra

diu

s ar

e co

ngr

uen

t.al

way

s

2.S

upp

ly t

he

mis

sin

g ra

tios

.

a.If

th

e ra

tio

of t

he

diam

eter

s of

tw

o sp

her

es i

s 3:

1,th

en t

he

rati

o of

th

eir

surf

ace

area

s

is

,an

d th

e ra

tio

of t

hei

r vo

lum

es i

s .

b.

If t

he

rati

o of

th

e ra

dii

of t

wo

hem

isph

eres

is

2:5,

then

th

e ra

tio

of t

hei

r su

rfac

e ar

eas

is

,an

d th

e ra

tio

of t

hei

r vo

lum

es i

s .

c.If

tw

o co

nes

are

sim

ilar

an

d th

e ra

tio

of t

hei

r h

eigh

ts i

s �4 3� ,

then

th

e ra

tio

of t

hei

r

volu

mes

is

,an

d th

e ra

tio

of t

hei

r su

rfac

e ar

eas

is

.

d.

If t

wo

cyli

nde

rs a

re s

imil

ar a

nd

the

rati

o of

th

eir

surf

ace

area

s is

100

:49,

then

th

e

rati

o of

th

e ra

dii

of t

hei

r ba

ses

is

,an

d th

e ra

tio

of t

hei

r vo

lum

es i

s

.

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al c

once

pt i

s to

rel

ate

it t

o so

met

hin

g yo

ual

read

y kn

ow.H

ow c

an w

hat

you

kn

ow a

bou

t th

e u

nit

s u

sed

to m

easu

re l

engt

hs,

area

s,an

d vo

lum

es h

elp

you

to

rem

embe

r th

e th

eore

m a

bou

t th

e ra

tios

of

surf

ace

area

s an

dvo

lum

es o

f si

mil

ar s

olid

s?S

amp

le a

nsw

er:

Len

gth

s ar

e m

easu

red

in li

nea

ru

nit

s,su

rfac

e ar

eas

in s

qu

are

un

its,

and

vo

lum

es in

cu

bic

un

its.

Take

th

esc

ale

fact

or,

wh

ich

is t

he

rati

o o

f lin

ear

mea

sure

men

ts in

th

e so

lids,

and

squ

are

it t

o g

et t

he

rati

o o

f th

eir

surf

ace

area

s o

r cu

be

it t

o g

et t

he

rati

oo

f th

eir

volu

mes

.

1000

:343

10:7

�1 96 ��6 24 7�

8:12

54:

25

27:1

9:1

©G

lenc

oe/M

cGra

w-H

ill74

6G

lenc

oe G

eom

etry

Co

ng

ruen

t an

d S

imila

r S

olid

s

Det

erm

ine

wh

eth

er e

ach

pai

r of

sol

ids

is s

imil

ar,

con

gru

ent,

or n

eith

er.

1.2.

nei

ther

sim

ilar

3.4.

con

gru

ent

sim

ilar

Th

e tw

o re

ctan

gula

r p

rism

s sh

own

at

the

righ

t ar

e si

mil

ar.

5.F

ind

the

rati

o of

th

e pe

rim

eter

s of

th

e ba

ses.

7:5

6.W

hat

is

the

rati

o of

th

e su

rfac

e ar

eas?

72:5

2o

r 49

:25

7.S

upp

ose

the

volu

me

of t

he

smal

ler

pris

m i

s 60

in

3 .F

ind

the

volu

me

of t

he

larg

er p

rism

.16

4.64

in3

Det

erm

ine

wh

eth

er e

ach

sta

tem

ent

is t

rue

or f

als

e.If

th

e st

atem

ent

is f

alse

,rew

rite

it

so t

hat

it

is t

rue.

8.If

tw

o cy

lin

ders

are

sim

ilar

,th

en t

hei

r vo

lum

es a

re e

qual

.Fa

lse;

if t

wo

cyl

ind

ers

are

con

gru

ent,

then

th

eir

volu

mes

are

eq

ual

.

9.D

oubl

ing

the

hei

ght

of a

cyl

inde

r do

ubl

es t

he

volu

me.

tru

e

10.T

wo

soli

ds a

re c

ongr

uen

t if

th

ey h

ave

the

sam

e sh

ape.

Fals

e;tw

o s

olid

s ar

e si

mila

r if

th

ey h

ave

the

sam

e sh

ape.

7 in

.5

in.

24 y

d

12 y

d

12 y

d6

yd 8 yd

16 y

d12

m

3 m

3 m

3 m

3 m

3 m

3 m

4 m

4 m

4 m

4 m

4 m

10 m

48 m

16 m15

m

14 c

m

11 c

m

7 cm

7 cm

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-4

13-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Co

ord

inat

es in

Sp

ace

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-5

13-5

©G

lenc

oe/M

cGra

w-H

ill74

7G

lenc

oe G

eom

etry

Lesson 13-5

Gra

ph

So

lids

in S

pac

eIn

spa

ce,y

ou c

an d

escr

ibe

the

loca

tion

of

a po

int

usi

ng

an o

rder

ed t

rip

leof

rea

ln

um

bers

.Th

e x-

,y-,

and

z-ax

es a

re p

erpe

ndi

cula

r to

ea

ch o

ther

,an

d th

e co

ordi

nat

es f

or p

oin

t P

are

the

orde

red

trip

le (

�4,

6,5)

.A r

ecta

ngu

lar

pris

m c

an b

edr

awn

to

show

per

spec

tive

.

Gra

ph

th

e re

ctan

gula

r so

lid

th

at

con

tain

s th

e or

der

ed t

rip

le (

2,1,

�2)

an

d t

he

orig

in.L

abel

th

e co

ord

inat

es o

f ea

ch v

erte

x.•

Plo

t th

e x-

coor

din

ate

firs

t.D

raw

a s

olid

seg

men

t fr

om t

he

orig

in 2

un

its

in t

he

posi

tive

dir

ecti

on.

•P

lot

the

y-co

ordi

nat

e n

ext.

Dra

w a

sol

id s

egm

ent

1 u

nit

in

th

e po

siti

ve d

irec

tion

.•

Plo

t th

e z-

coor

din

ate

nex

t.D

raw

a s

olid

seg

men

t 2

un

its

in t

he

neg

ativ

e di

rect

ion

.•

Dra

w t

he

rect

angu

lar

pris

m,u

sin

g do

tted

lin

es f

or

hid

den

edg

es o

f th

e pr

ism

.•

Lab

el t

he

coor

din

ates

of

each

ver

tex.

Gra

ph

th

e re

ctan

gula

r so

lid

th

at c

onta

ins

the

give

n p

oin

t an

d t

he

orig

in a

sve

rtic

es.L

abel

th

e co

ord

inat

es o

f ea

ch v

erte

x.

1.A

(2,1

,3)

2.G

(�1,

2,3)

3.P

(�2,

1,�

1)4.

T(�

1,3,

2)

y

x

z

( 0, 0

, 0)

( 0, 3

, 0)

( �1,

3, 0

)

( 0, 0

, 2)

( 0, 3

, 2)

( �1,

0, 2

)

( �1,

0, 0

)

T( �

1, 3

, 2)

y

x

z

( 0, 0

, 0)

( 0, 1

, 0)

P( �

2, 1

, �1)

( �2,

0, �

1)( �

2, 1

, 0)

( �2,

0, 0

)

( 0, 0

, �1)

( 0, 1

, �1)

y

x

z

( 0, 0

, 0)

( 0, 0

, 3)

( �1,

2, 0

)( �

1, 0

, 0)

( �1,

0, 3

)G

( �1,

2, 3

)

( 0, 2

, 0)

( 0, 2

, 3)

y

x

z( 0

, 0, 3

)

( 0, 0

, 0)

( 2, 0

, 3)

( 2, 0

, 0)

( 2, 1

, 0)

( 0, 1

, 0)

( 0, 1

, 3)

A( 2

, 1, 3

)

y

x

z

( 0, 0

, 0)

( 0, 1

, 0)

( 0, 1

, �2)

( 2, 1

, �2)

( 2, 0

, �2)

( 0, 0

, �2)

( 2, 0

, 0)

( 2, 1

, 0)

y

x

z O

P( �

4, 6

, 5)

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill74

8G

lenc

oe G

eom

etry

Dis

tan

ce a

nd

Mid

po

int

Form

ula

sYo

u c

an e

xten

d th

e D

ista

nce

For

mu

la a

nd

the

Mid

poin

t F

orm

ula

to

thre

e di

men

sion

s to

fin

d th

e di

stan

ce b

etw

een

tw

o po

ints

in

spa

ce

and

to f

ind

the

mid

poin

t of

th

e se

gmen

t co

nn

ecti

ng

two

poin

ts.

Dis

tan

ce F

orm

ula

Giv

en t

wo

poin

ts A

(x1,

y1,

z1)

and

B(x

2, y

2, z

2) in

spa

ce,

the

dist

ance

bet

wee

n

in S

pac

eA

and

Bis

giv

en b

y A

B�

�(x

1�

�x 2

)2�

�(y

1�

�y 2

)2�

�(z

1�

�z 2

)2�

.

Mid

po

int

Fo

rmu

laG

iven

tw

o po

ints

A(x

1, y

1, z

1) a

nd B

(x2,

y2,

z2)

in s

pace

, th

e m

idpo

int

of A�

B�is

in S

pac

eat

��x 1� 2

x 2�

, �y 1

� 2y 2

�, �z 1

� 2z 2

��.

Det

erm

ine

the

dis

tan

ce b

etw

een

A(3

,2,�

5) a

nd

B(�

4,6,

9).

Th

en d

eter

min

e th

e co

ord

inat

es o

f th

e m

idp

oin

t of

A �B�

.

AB

��

(x1

��

x 2)2

��

(y1

��

y 2)2

��

(z1

��

z 2)2

��

�(3

�(

��

4))2

��

(2 �

�6)

2�

�(�

5 �

�9)

2�

��

72�

(�

�4)

2�

�(�

14�

)2 ��

�49

��

16 �

�19

6�

�16

.2

mid

poin

t of

A�B�

��x 1

� 2x 2

�,�

y 1� 2

y 2�

,�z 1

� 2z 2

��

��3

�

2(�4)

�,�

2� 2

6�

,��

5 2�9

��

�(�

0.5,

4,2)

Det

erm

ine

the

dis

tan

ce b

etw

een

eac

h p

air

of p

oin

ts.T

hen

det

erm

ine

the

coor

din

ates

of

the

mid

poi

nt

Mof

th

e se

gmen

t jo

inin

g th

e p

air

of p

oin

ts.

1.A

(0,7

,�4)

an

d B

(�2,

8,3)

2.C

(�7,

6,5)

an

d D

(10,

2,�

5)

AB

��

54��

7.3

;M

��1,

�1 25 �,�

�1 2� �C

D�

�40

5�

�20

.1;

M��3 2� ,

4,0 �

3.E

(3,1

,�2)

an

d F

(�2,

3,4)

4.G

(�4,

1,1)

an

d H

(0,2

,�1)

EF

��

65��

8.1

;M

��1 2� ,2,

1 �G

H�

�21�

�4.

6;M

��2,

�3 2� ,0 �

5.J

(6,1

,�2)

an

d K

(�1,

�2,

1)6.

L(�

5,0,

�3)

an

d N

(0,0

,�4)

JK�

�67�

�8.

2 ;

M��5 2� ,

��1 2� ,

��1 2� �

LN

��

26��

5.1;

M��

�5 2� ,0,

��7 2� �

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Co

ord

inat

es in

Sp

ace

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-5

13-5

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Co

ord

inat

es in

Sp

ace

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-5

13-5

©G

lenc

oe/M

cGra

w-H

ill74

9G

lenc

oe G

eom

etry

Lesson 13-5

Gra

ph

th

e re

ctan

gula

r so

lid

th

at c

onta

ins

the

give

n p

oin

t an

d t

he

orig

in a

sve

rtic

es.L

abel

th

e co

ord

inat

es o

f ea

ch v

erte

x.

1.A

(�5,

3,2)

2.H

(3,2

,5)

3.D

ilat

e th

e pr

ism

by

a sc

ale

fact

or o

f 2.

Gra

ph t

he

imag

e u

nde

r th

e di

lati

on.

A�(

�4,

6,2)

,B�(

�4,

0,2)

,C�(

0,0,

2),D

�(0,

6,2)

,E

�(�

4,6,

0),F

�(�

4,0,

0),G

�(0,

0,0)

,H�(

0,6,

0)

Det

erm

ine

the

dis

tan

ce b

etw

een

eac

h p

air

of p

oin

ts.T

hen

det

erm

ine

the

coor

din

ates

of

the

mid

poi

nt

Mof

th

e se

gmen

t jo

inin

g th

e p

air

of p

oin

ts.

4.R

(2,1

,0)

and

S(3

,3,4

)5.

Q(5

,0,�

2) a

nd

T(2

,3,2

)

RS

��

21�;��5 2� ,

2,2 �

QT

��

34�;��7 2� ,

�3 2� ,0 �

6.A

(�4,

1,6)

an

d B

(�1,

0,4)

7.J

(0,5

,1)

and

K(4

,�3,

2)

AB

��

14�;��

�5 2� ,�1 2� ,

5 �JK

�9;

�2,1,

�3 2� �

y

x

z

A�

B�

C�

D�

E�

F�

G�

H�

y

x

z

AB

C

GH

ED

F

y

x

z

N( 0

, 0, 0

)

M( 3

, 0, 0

)L

( 3, 2

, 0)

P( 0

, 2, 0

)

K( 0

, 2, 5

)J(

0, 0

, 5)

I(3,

0, 5

)

H( 3

, 2, 5

)

y

x

zA

( �5,

3, 2

)

E( �

5, 3

, 0)

B( �

5, 0

, 2)

C( 0

, 0, 2

)D

( 0, 3

, 2)

H( 0

, 3, 0

)

G( 0

, 0, 0

)

F( �

5, 0

, 0)

©G

lenc

oe/M

cGra

w-H

ill75

0G

lenc

oe G

eom

etry

Gra

ph

th

e re

ctan

gula

r so

lid

th

at c

onta

ins

the

give

n p

oin

t an

d t

he

orig

in a

sve

rtic

es.L

abel

th

e co

ord

inat

es o

f ea

ch v

erte

x.

1.E

(4,6

,�2)

2.R

(�3,

�5,

4)

Det

erm

ine

the

dis

tan

ce b

etw

een

eac

h p

air

of p

oin

ts.T

hen

det

erm

ine

the

coor

din

ates

of

the

mid

poi

nt

Mof

th

e se

gmen

t jo

inin

g th

e p

air

of p

oin

ts.

3.Y

(�5,

1,2)

an

d Z

(3,�

3,1)

4.E

(4,2

,0)

and

F(3

,2,�

2)

YZ

�9;

��1,

�1,

�3 2� �E

F�

�5�;

��7 2� ,2,

�1 �

5.B

(�2,

�2,

�3)

an

d C

(1,�

3,0)

6.H

(2,0

,�3)

an

d I(

4,�

1,5)

BC

��

19�;��

�1 2� ,�

�5 2� ,�

�3 2� �H

I��

69�;�3,

��1 2� ,

1 �

7.A

NIM

ATI

ON

Der

ek w

ants

to

anim

ate

an i

mag

e fo

r h

is s

cien

ce p

rese

nta

tion

by

mov

ing

it f

rom

on

e po

siti

on t

o an

oth

er.T

he

mes

h o

f th

e im

age

is a

rec

tan

gula

r pr

ism

wit

hco

ordi

nate

s A

(�3,

2,3)

,B(�

3,0,

3),C

(0,0

,3),

D(0

,2,3

),E

(�3,

2,0)

,F(�

3,0,

0),G

(0,0

,0),

and

H(0

,2,0

).F

ind

the

coor

dina

tes

of t

he m

esh

afte

r th

e tr

ansl

atio

n (x

,y,z

) →

(x�

7,y,

z).

Gra

ph b

oth

th

e pr

eim

age

and

imag

e of

th

e m

esh

.A

�(�

10,2

,3),

B�(

�10

,0,3

),C

�(�

7,0,

3),

D�(

�7,

2,3)

,E�(

�10

,2,0

),F

�(�

10,0

,0),

G�(

�7,

0,0)

,H�(

�7,

2,0)

y

x

zA

�B

�

C�

D�

E�

F�

G�

H�

AB

CD

EF

GH

y

x

z

X( 0

, 0, 0

)

Y( 0

, �5,

0)

V( �

3, �

5, 0

)R( �

3, �

5, 4

)S

( �3,

0, 4

)

T( 0

, 0, 4

)U

( 0, �

5, 4

)

W( �

3, 0

, 0)

y

x

z

K( 0

, 0, 0

)

J(4,

0, 0

)I(

4, 6

, 0)

L( 0

, 6, 0

)

H( 0

, 6, �

2)

E( 4

, 6, �

2)F

( 4, 0

, �2)

G( 0

, 0, �

2)

Pra

ctic

e (

Ave

rag

e)

Co

ord

inat

es in

Sp

ace

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-5

13-5



Readin

g t

o L

earn

Math

em

ati

csC

oo

rdin

ates

in S

pac

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-5

13-5

©G

lenc

oe/M

cGra

w-H

ill75

1G

lenc

oe G

eom

etry

Lesson 13-5

Pre-

Act

ivit

yH

ow i

s th

ree-

dim

ensi

onal

gra

ph

ing

use

d i

n c

omp

ute

r an

imat

ion

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 13

-5 a

t th

e to

p of

pag

e 71

4 in

you

r te

xtbo

ok.

Wh

y w

ould

a m

esh

be

crea

ted

firs

t?S

amp

le a

nsw

er:

A m

esh

is a

no

utl

ine

that

th

e an

imat

or

wo

uld

use

fir

st li

ke a

ske

tch

bef

ore

ren

der

ing

a f

inal

imag

e.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.M

atch

eac

h p

oin

t fr

om t

he

firs

t co

lum

n

wit

h i

ts c

oord

inat

es f

rom

th

e se

con

d co

lum

n.

a.A

iiii.

(3,0

,0)

b.

Bvi

ii.(

3,0,

�4)

c.O

vii

i.(3

,�2,

0)

d.

Ji

iv.

(3,�

2,�

4)

e.H

ivv.

(0,0

,0)

f.K

iivi

.(0,

�2,

0)

g.T

viii

vii.

(0,�

2,�

4)

h.

Rvi

ivi

ii.(

0,0,

�4)

2.W

hic

h o

f th

e fo

llow

ing

expr

essi

ons

give

th

e di

stan

ce b

etw

een

th

e po

ints

at

(4,�

1,�

5)an

d (�

3,2,

�9)

?A

,E,H

A.

�72

�(

��

3)2

��

42 �B

.�12

�1

�2

�(�

�14

)2�

C.

�22

�2

�2

�42

�D

. ��1 2� ,�1 2� ,

�7 �

E.

�(�

3 �

�4)

2�

�(�

1 �

�2)

2�

�(�

9 �

�5)

2�

F.�

24�

G.�

(�3

��

4)2

��

[2 �

�(�

1)]2

��

[��

9 �

(��

5)]2

�H

.�74�

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

new

mat

hem

atic

al f

orm

ula

s is

to

rela

te t

hem

to

ones

you

alre

ady

know

.How

can

you

use

you

r kn

owle

dge

of t

he

Dis

tan

ce a

nd

Mid

poin

t F

orm

ula

sin

tw

o di

men

sion

s to

rem

embe

r th

e fo

rmu

las

in t

hre

e di

men

sion

s?S

amp

le a

nsw

er:

Sta

rt w

ith

th

e fo

rmu

las

for

two

dim

ensi

on

s.A

dd

a t

hir

d t

erm

un

der

th

era

dic

al in

th

e D

ista

nce

Fo

rmu

la a

nd

a t

hir

d c

oo

rdin

ate

in t

he

Mid

po

int

Fo

rmu

la t

hat

is ju

st li

ke t

he

oth

er t

wo

exc

ept

that

th

e va

riab

le is

zra

ther

than

xo

r y.

y

x

z

A

KBO

R

H

T

J

©G

lenc

oe/M

cGra

w-H

ill75

2G

lenc

oe G

eom

etry

Pla

nes

an

d C

ylin

dri

cal S

urf

aces

Con

side

r th

e po

ints

(x,

y,z)

in

spa

ce w

hos

e co

ordi

nat

es s

atis

fy t

he

equ

atio

n z

�1.

Sin

ce x

an

dy

do n

ot o

ccu

r in

th

e eq

uat

ion

,an

y po

int

wit

h i

ts z

-coo

rdin

ate

equ

al t

o 1

has

coo

rdin

ates

th

at s

atis

fy t

he

equ

atio

n.T

hes

e ar

e th

e po

ints

in

th

e pl

ane

1 u

nit

abo

ve t

he

xy-p

lan

e.T

his

pl

ane

is p

erpe

ndi

cula

r to

th

e z-

axis

at

(0,0

,1).

Nex

t co

nsi

der

the

poin

ts (

x,y,

z) w

hos

e co

ordi

nat

es s

atis

fy x

2�

y2�

16.I

n t

he

xy-p

lan

e,al

l po

ints

on

th

e ci

rcle

wit

h c

ente

r (0

,0,0

) an

dra

diu

s 4

hav

e co

ordi

nat

es t

hat

sat

isfy

th

e eq

uat

ion

.In

th

e pl

ane

perp

endi

cula

r to

th

e z-

axis

at

(0,0

,k),

the

poin

ts t

hat

sat

isfy

th

eeq

uat

ion

are

th

ose

on t

he

circ

le w

ith

cen

ter

(0,0

,k)

and

radi

us

4.T

he

grap

h i

n s

pace

of

x2�

y2�

16 i

s an

in

fin

ite

cyli

ndr

ical

su

rfac

e w

hos

e ax

is i

s th

e z-

axis

an

d w

hos

e ra

diu

s is

4.

Des

crib

e th

e gr

aph

in

sp

ace

of e

ach

eq

uat

ion

.You

may

fin

d i

t h

elp

ful

to m

ake

sket

ches

on

a s

epar

ate

shee

t.

1.x

�5

the

pla

ne

per

pen

dic

ula

r to

th

e x-

axis

at

(5,0

,0)

2.y

��

2th

e p

lan

e p

erp

end

icu

lar

to t

he

y-ax

is a

t (0

,�2,

0)

3.x

�y

�7

the

pla

ne

par

alle

l to

th

e z-

axis

an

d c

on

tain

ing

th

e lin

e th

rou

gh

(0,7

,0)

and

(7,

0,0)

4.z2

�y2

�25

the

infi

nit

e cy

lind

rica

l su

rfac

e w

ho

se a

xis

is t

he

x-ax

is a

nd

wh

ose

rad

ius

is 5

5.(x

�2)

2�

(y�

5)2

�1

the

infin

ite c

ylin

dri

cals

urf

ace

wh

ose

axi

s is

th

e lin

ep

aral

lel t

o t

he

z-ax

is a

nd

pas

sin

g t

hro

ug

h (

2,5,

0) a

nd

wh

ose

rad

ius

is 1

6.x2

�y2

�z2

�0

the

po

int

(0,0

,0)

z

y

x

O

(0, 0

, k)

plan

e fo

r z �

k

z

y

x

O

(0, 0

, 1)

N

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

13-5

13-5



Chapter 13 Assessment Answer Key Form 1 Form 2APage 753 Page 754 Page 755

(continued on the next page)

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

A

B

D

C

C

A

B

D

A

B

B

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

B

D

B

D

B

D

A

D

C

a cylinder with a3-cm radius

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

C

C

A

C

C

D

B

B

A

D

A


Chapter 13 Assessment Answer KeyForm 2A (continued) Form 2BPage 756 Page 757 Page 758

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

D

B

C

D

A

D

D

B

32.4 in3

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

D

C

C

A

D

B

D

B

B

B

A

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

B

A

C

B

D

D

A

C

A

326.7 cm3


Chapter 13 Assessment Answer KeyForm 2CPage 759 Page 760

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

432 in3

2 ft

7690.6 ft3

1885.0 in3

270�3� ft3

7.2 in.

18 times

103.2 ft3

208.8 in3

2145 in3

1393.0 in3

968.7 cm3

13.

14.

15.

16.

17.

18.

19.

20.

B:

always

neither

612.5 m2

4:9

D�(15, �5, �3)

(1, 5, 7)

3�2�

314.2 in3

y

x

z

A


Chapter 13 Assessment Answer KeyForm 2DPage 761 Page 762

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

144 cm3

576 in3

923.6 cm3

3 cm

2057 in3

36 in2

12 times

100.5 cm3

106.4 cm3

7238 ft3

113.1 in3

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

38.7 in3

always

similar

1562.5 ft3

512:729

X�(�28, 8, 20)

14 units

(�8, 6, 4)

��14

�, �241�, ��

121��

y

x

z

A


Chapter 13 Assessment Answer KeyForm 3Page 763 Page 764

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

125 m3

184 cm3

114�3� in3

8 m

No, it willoverflow.

148.4 in3 � 96 in3

212.1 cm3

266.9 cm3

121.5 in3

263.9 in3

138.2 cm3

75,266.3 cm3

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

64 times

Yes; �32

3�� 12�.

similar

1:3�4�

45 in.

C(3, �9, 10)

�14� units

118.0 mi

7104 mL

y

x

z


Chapter 13 Assessment Answer KeyPage 765, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofcongruent solids, similar solids, volume, ratios of surfaceareas and volumes, ordered triples, graphing points inspace, midpoints, and distances in three dimensions.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Figures and graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of congruentsolids, similar solids, volume, ratios of surface areas andvolumes, ordered triples, graphing points in space,midpoints, and distances in three dimensions.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures and graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofcongruent solids, similar solids, volume, ratios of surfaceareas and volumes, ordered triples, graphing points inspace, midpoints, and distances in three dimensions.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Figures and graphs are mostly accurate.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work shown to substantiate the

final computation.• Figures and graphs may be accurate but lack detail or

explanation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts ofcongruent solids, similar solids, volume, ratios of surfaceareas and volumes, ordered triples, graphing points inspace, midpoints, and distances in three dimensions.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Figures and graphs are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations


An

swer

s

Chapter 13 Assessment Answer KeyPage 765, Open-Ended Assessment

Sample Answers

1. Sample coordinates:A(8, 0, 0), B(8, 16, 0), C(0, 16, 0),D(0, 0, 0), E(8, 0, 12), F(8, 16, 12),G(0, 16, 12), H(0, 0, 12);A�(2, 0, 0), B�(2, 4, 0), C�(0, 4, 0),D�(0, 0, 0), E�(2, 0, 3), F �(2, 4, 3),G�(0, 4, 3), H�(0, 0, 3)

2. The volume of the cylinder is �r2 � 2r.

The volume of the hemisphere is �2�3r3�.

The volume of the cone is �r2 � �3r

�.

Therefore, the total volume is

2�r3 � �2�

3r3� � �

�3r3� or 3�r3.

3. The first cylinder could have a radius of 3,a height of 4, a volume of 36�, and asurface area of 42�. The second cylindercould have a radius of 1, a height of 30, avolume of 30�, and a surface area of 62�.

4.

The volume of the pyramid is �163� 12� or

64 cubic units and the volume of theprism is 4 � 4 � 4 or 64 cubic units.

5. The formula for the volume of a prism isV � Bh. Since the base of a cylinder is acircle, its area is �r2. If you substitute�r2 for B in the V � Bh formula, youobtain V � �r2h for the volume of acylinder.

6. A sample pair of points are A(2, 3, 4) andC(4, 5, 6). Accept any pair of coordinatesA(x1, y1, z1) and A(x2, y2, z2) in which x1 � x2 � 6, y1 � y2 � 8, and z1 � z2 � 10.

4 in.4 in.4 in.

12 in.

4 in.

4 in.

y

x

z

44

4

In addition to the scoring rubric found on page A22, the following sample answers may be used as guidance in evaluating open-ended assessment items.


Chapter 13 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 766 Page 767 Page 768

Quiz 2Page 767

Quiz 4Page 768

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

false,similar solids

false,congruent solids

false,ordered triple

true

false, V � ��r

3

2h�

false, V � �4�

3r 3�

true

true

false, V � �r 2h

true

a3:b3

��x1 �

2x2�, �

y1 �

2y2�, �

z1 �

2z2��

1.

2.

3.

4.

5.

1152 ft3

2 cm

2827.4 in3

156 cm3

1005.3 cm3

1.

2.

3.

4.

5.

8 times

261.8 cm3

904.8 mm3

160.1 mm3

C

1.

2.

3.

4.

5.

neither

congruent

4:9

0.84 in.

false

1.

2.

3.

4.

5.

A�(5, �3, 5),B�(6, �2, 5),C�(2, �1, 5),D�(5, �2, 8)

�2, �52

�, �129��

�74�

C(14, �2, 11)

y

x

z

B


Chapter 13 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 769 Page 770

An

swer

s

Part I

Part II

6.

7.

8.

9.

10.

6.4 cm3

960 cm3

7.5 in3

1150.3 cm3

32�3� in3

1.

2.

3.

4.

5.

A

D

C

B

B

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

6

�183�

isosceles

72° and 108°

30 cm by 18 cm

38

(x � 1)2 �

(y � 2)2 � 16

936 cm2

6; All faces are squares.

315 cm2

1357.2 m3

168 in3


Chapter 13 Assessment Answer KeyStandardized Test Practice

Page 771 Page 772

1.

2.

3.

4.

5.

6.

7.

8. E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D9. 10.

11.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

12.

13.

14.

15.

16.

10.25

126 m2

175.9 in2

1526.8 ft3

(0.5, 5.5, �6)

4 2 5

3


Chapter 13 Assessment Answer KeyUnit 4 Test/Review (Ch. 11–13)

Page 773 Page 774

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

parallelogram;70 units2

9.5 cm

324� � 486�3�in2

33 units2

0.44

triangular prism;bases: �ABC

and �DEF

1009.4 units2

2

129.8 in2

8.5 m

11.

12.

13.

14.

15.

16.

17.

18.

127.5 ft2

181.2 in2

2035.8 cm2

102.4 ft3

1900.7 in3

cylinder

similar

y

x

z

(0, 0, 0)

(0, 0, �3)

(�1, �4, 0)(�1, 0, 0)

(�1, 0, �3)

M(�1, �4, �3)

(0, �4, 0)

(0, �4, �3)


Chapter 13 Assessment Answer KeySecond Semester Test (Ch. 8–13)

Page 775 Page 776

1.

2.

3.

4.

5.

6.

7.

8.

A

D

B

B

C

A

B

C

9.

10.

11.

12.

13.

14.

15.

16.

17.

A

C

C

A

C

A

D

B

D


Chapter 13 Assessment Answer KeySecond Semester Test (Ch. 8–13)

Page 777 Page 778

An

swer

s

18.

19.

20.

21.

22.

23.

24.

25.

26.

1800; 30

x � 12; y � 4

Rhombus,rectangle, square;consecutive sides

are ⊥; all sides are �.

(b � c, 0)

25.5

resultant speed:

� 20.4 mph;direction: � 11.3°west of due south

m�1 � 68, m�2 �112, m�3 � 68,

m�4 � 90

34

D�

E�F�

G�

D

EF

G

x

y

O

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

25.5

12.5

30

49.7 mm2

about 0.41

1080 units2

104 cm

sphere

barrel

120 cm3

�453�


Chapter 13 Assessment Answer KeyFinal Test

Page 779 Page 780

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

D

A

B

D

C

B

A

C

D

A

11.

12.

13.

14.

15.

16.

17.

18.

D

A

A

D

B

B

C

B



Page 781 Page 782

An

swer

s

19.

20.

21.

22.

23.

24.

25.

26.

A

D

C

C

B

A

C

B

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

H

22; 98

hypothesis: a circle

has a radius of r ;conclusion: its

circumference is 2�r

always

60

22; 18

A�G� � E�H�, C�A� � C�E�,�CGH � �CHG

Given

If 2 � of a � are �,then the sidesopposite those

�s are �.

�AGC � �EHC

SSS

centroid

x

y

O

K(3, 4)



Page 783 Page 784

40.

41.

42.

43.

44.

45.

46.

47.

48.

y � 3

5; 35

7

12; 9; 25

52�2� or �73.5;52

m�E � 52, m�F �

127, m�G � 99,m�H � 82

13; 11

5; yes

1 → 2: (x, y) →(x � 3, y � 7);2 → 3: (x, y) →(x � 4, y � 4)

49.

50.

51.

52.

53.

54.

55.

56.

14

112; 68

1387.7 ft2

400�; �0.42

554.1 cm2

hemisphere

�115�;(0.5, �0.5, �10.5)

Chapter 13 Resource Masters - Math Problem Solving©Glencoe/McGraw-Hill iv Glencoe Geometry...

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Transcript of Chapter 13 Resource Masters - Math Problem Solving©Glencoe/McGraw-Hill iv Glencoe Geometry...