Area · 2017. 4. 4. · A quadrilateral is a closed four-sided geometric figure. Opposite sides are...

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817

Area

Sleds, diamonds, and

deltas. Those are the most popular kinds of kites. But there are also rollers,

dragons, doperos, and rokkakus. Kite flying is a popular activity around

the world.

13.1 TheLanguageofGeometrySketching, Drawing, Naming, and Sorting

Basic Geometric Figures ............................................. 819

13.2WeavingarugArea and Perimeter of Rectangles and Squares ...........833

13.3 BoundaryLinesArea of Parallelograms and Triangles ......................... 847

13.4 TheKeystoneEffectArea of Trapezoids ..................................................... 863

13.5 GoFlyaKiteArea of Rhombi and Kites ...........................................875

13.6 StreetSignsArea of Regular Polygons .......................................... 883

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818 • Chapter 13 Area

13.1 Sketching, Drawing, Naming, and Sorting Basic Geometric Figures • 819

TheLanguageofGeometrySketching, Drawing, Naming, and Sorting Basic Geometric Figures

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Learning GoalsIn this lesson, you will:

Classify geometric figures as polygons, triangles,

quadrilaterals, pentagons,

hexagons, heptagons,

octagons, nonagons, and

decagons.

Define consecutive sides and opposite sides.

Sort polygons into categories.

Key Terms protractor

compass

straightedge

sketch

draw

construct

triangle

equilateral triangle

isosceles triangle

scalene triangle

equiangular triangle

acute triangle

right triangle

obtuse triangle

quadrilateral

opposite sides

consecutive sides

square

rectangle

rhombus

parallelogram

kite

trapezoid

isosceles trapezoid

polygon

regular polygon

irregular polygon

pentagon

hexagon

heptagon

octagon

nonagon

decagon

When you sketch a geometric figure, you create the figure without the use of tools. When you draw a geometric figure, you create the figure with the use of

tools. A drawing is more accurate than a sketch. Any tools can be used, such as

rulers, straightedges, compasses, protractors, etc.

Look at the figure shown.

Make a sketch of the figure, and then make a drawing of

the figure. Compare your sketches and drawings to your

classmates. What do you notice? Were the sketches or

drawings more exact copies of the figure shown?


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Problem 1 Triangles

Producing pictures, sketches, diagrams, and drawings of figures is a very important part of

geometry. Many tools can be used to create geometric figures. Some tools, such as a ruler

or a protractor, are classified as measuring tools. A protractor can be used to approximate

the measure of an angle. A compass is a tool used to create arcs and circles. A

straightedge is a ruler with no numbers. It is important to know when to use each tool.

● When you sketch a geometric figure, the figure is created without the use of tools.

● When you draw a geometric figure, the figure is created with the use of tools such as a

ruler, a straightedge, a compass, or a protractor. A drawing is generally more accurate

than a sketch.

● When you construct a geometric figure, the figure is created using only a compass and

a straightedge.

You have already worked with basic geometric figures such as points, lines, rays, planes,

line segments, and angles. These basic figures can be used to build more complex

geometric figures. A triangle is the simplest closed three-sided geometric figure.

1. Sketch three different triangles.

2. How many line segments, angles, and vertices are needed to form a triangle?

A triangle is named using three capital letters representing the vertices, listed in a clockwise

or counterclockwise order. Triangle RAD can be written using symbols as nRAD. This is

read as “triangle RAD.” The triangle shown could be named nRAD, nADR, or nDRA.

R

A D

The root word “tri” means “three,” so triangle literally means “three angles.” A triangle is a

closed figure because it has a well-defined interior and exterior.

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3. Label the vertices of the triangles you sketched in Question 1, and then use symbols

to name each triangle.

4. Name the three sides and the three angles of nRAD.

Sides:

Angles:

Triangles are classified by their side lengths or by their angle measures.

The root word “equi” means “equal,” and the root word “lateral” means

“side.” An equilateraltriangle is a triangle with all sides congruent.

5. Draw an equilateral triangle. Label the drawing and include

measurements that verify it is an equilateral triangle.

An isoscelestriangle is a triangle with at least two congruent sides.

6. Draw an isosceles triangle. Label the drawing and include measurements that verify it

is an isosceles triangle.

Don't forget the

symbols when naming sides, like AD, and angles, like DAR.

So, an equilateral triangle is

also an isosceles triangle.”


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A scalenetriangle is a triangle with no congruent sides.

7. Draw a scalene triangle. Label the drawing and include measurements that verify it is

a scalene triangle.

An equiangulartriangle is a triangle with all angles congruent.

8. Draw an equiangular triangle. Label the drawing and include measurements that verify

it is an equiangular triangle.

9. How are equilateral and equiangular triangles related?

An acutetriangle is a triangle that has three angles that each

measure less than 90°.

10. Draw an acute triangle. Label the drawing and include

measurements that verify it is an acute triangle.

Time to get out your

protractor.

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A righttriangle is a triangle that has a right angle.

11. Draw a right triangle. Label the drawing and include measurements that verify it is a

right triangle.

An obtusetriangle is a triangle that has an angle measuring greater than 90°.

12. Draw an obtuse triangle. Label the drawing and include measurements that verify it is

an obtuse triangle.

Problem 2 Quadrilaterals

Aquadrilateral is a closed four-sided geometric figure.

Oppositesides are sides that do not share a common endpoint.

Consecutivesides are sides that share a common endpoint.

1. Sketch three different quadrilaterals.

2. How many line segments, angles, and vertices are needed to form a quadrilateral?

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A quadrilateral is named using four capital letters representing the vertices, listed in a

clockwise or counterclockwise order. The quadrilateral shown could be named

quadrilateral ABCD, quadrilateral ADCB, quadrilateral BCDA, and so on.

A

B

CD

3. What are three other names for quadrilateral ABCD not listed above?

4. Name the two pairs of opposite sides of quadrilateral ABCD.

5. Name the four pairs of consecutive sides of quadrilateral ABCD.

6. Name the four sides and the four angles of quadrilateral ABCD.

Sides:

Angles:

The root word “quad” means “four,” and the root word “lateral” means “side,” so

quadrilateral literally means “four sides.” A quadrilateral is a closed figure because it has a

well-defined interior and exterior.

7. Label the vertices of the quadrilaterals you sketched in Question 1, and then name

each quadrilateral.

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A square is a quadrilateral with all sides congruent and all angles congruent.

8. Draw a square. Label the drawing and include measurements.

A rectangle is a quadrilateral with opposite sides congruent and all angles congruent.

9. Draw a rectangle that is not a square. Label the drawing and include measurements.

A rhombus is a quadrilateral with all sides congruent. The plural of

rhombus is rhombi.

10. Draw a rhombus that is not a square. Label the drawing and

include measurements.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

11. Draw a parallelogram that is not a rectangle or a rhombus. Label the drawing and

include measurements.

So, are all squares also

rectangles? Or, are all rectangles also

squares?

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A kite is a quadrilateral with two pairs of consecutive congruent sides with opposite sides

that are not congruent.

12. Draw a kite. Label the drawing and include measurements.

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

13. Draw a trapezoid. Label the drawing and include measurements.

An isoscelestrapezoid is a trapezoid whose non-parallel sides are congruent.

14. Draw an isosceles trapezoid. Label the drawing and include measurements.

Problem 3 Polygons

Triangles and quadrilaterals are examples of geometric figures with many sides. The root

word “poly” means “many” and the root word “gon” means “side.” A polygon is a closed

figure that is formed by joining three or more line segments at their endpoints. A regular

polygon is a polygon with all sides congruent and all angles congruent. An irregular

polygon is a polygon that is not regular. Polygons are named using capital letters

representing the vertices, listed in a clockwise or counterclockwise order.

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The root word “penta” means “five,” so pentagon literally means “five sides.” A pentagon

is a five-sided polygon.

1. Sketch a pentagon. Label the vertices and name the pentagon.

The root word “hexa” means “six,” so hexagon literally means “six sides.” A hexagon is a

six-sided polygon.

2. Sketch a hexagon. Label the vertices and name the hexagon.

The root word “hepta” means “seven,” so heptagon literally means “seven sides.” A

heptagon is a seven-sided polygon.

3. Sketch a heptagon. Label the vertices and name the heptagon.

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The root word “octa” means “eight,” so octagon literally means “eight sides.” An octagon

is an eight-sided polygon.

4. Sketch an octagon. Label the vertices and name the octagon.

The root word “nona” means “nine,” so nonagon literally means “nine sides.” A nonagon

is a nine-sided polygon.

5. Sketch a nonagon. Label the vertices and name the nonagon.

The root word “deca” means “ten,” so decagon literally means “ten sides.” A decagon is

a ten-sided polygon.

6. Sketch a decagon. Label the vertices and name the decagon.

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Problem 4 Sort Activity

1. Cut out each shape shown in the diagram. How would you sort these shapes?

AB

C

F

G

H

I

J

LO

M

K

E

D

Q

R

W

PN

V

U

T

S

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2. Tyler sorted all of the figures shown into two groups. One group contained all of the

figures with at least one pair of perpendicular sides. The second group contained all

of the figures that did not have at least one pair of perpendicular sides. Which figures

were in Tyler’s groups?

Group 1

Figures that have at least one pair of

perpendicular sides:

Group 2

Figures that do not have at least one

pair of perpendicular sides:

3. Molly sorted all of the figures shown into two groups. What characteristics describe

each of her two groups?

Group 1

A, E, F, G, H, I, J, K, L, M, N, O, P, T,

V, W

Group 2

B, C, D, Q, R, S, U

4. Carefully analyze the cut out figures. Think of a few characteristics different from the

ones Tyler and Molly used in the last two questions. Sort the figures into groups using

these characteristics. Explain your reasoning.

5. Compare the characteristics you used to those of your

classmates. What different characteristics did your

classmates use to sort the figures?

Be prepared to share your solutions and methods.

How are the figures in each of your groups similar?


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13.2 Area and Perimeter of Rectangles and Squares • 833


Calculate the area of rectangles and squares.

Calculate the perimeter of rectangles and squares.

Write a formula for the perimeter and area of a rectangle and a square.

Determine the effect of altering the dimensions of a rectangle or a square on the perimeter and area.

Calculate the area of composite figures.

Carpets and rugs have been art forms in the Middle East for centuries. Its history spans before the religions of Islam and Christianity. What is even more

remarkable is that carpet making and rug weaving is still an important part of

the economy of many Middle Eastern countries. Many experts have estimated

that rug weaving brought in approximately $420 million dollars to the Iranian

economy in 2008. And rug weaving is quite a popular occupation. It is estimated

that there are 1.2 million rug weavers in Iran. With rugs and carpets still being

made by hand, do you think that carpets and rugs can be made by machine? Do

you think machine made Persian carpets and rugs will have the same quality of

handmade rugs?

WeavingaRugArea and Perimeter of Rectangles and Squares


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Problem 1 A Rectangular Rug

Tyson is currently creating rectangular-shaped rugs.

1. One rectangular rug is seven feet long and three feet wide. Draw a model of this rug on

the grid shown. Each square on the grid represents a square that is one foot long and

one foot wide.

2. What is the area of this rug? Explain your calculation.

3. What is the perimeter of this rug? Explain your calculation.

Are the units of measure

the same for area and perimeter?

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4. Six different rectangles are drawn on the grid shown. The letters A through F name

each rectangle.

A

B

C

D

EF

5. Each square on the grid represents a square that is one foot long and one foot wide.

Complete the table to show the length, width, area, and perimeter of each rectangle.

Rectangle Length (units)Width (units)

Perimeter (units)

Area (square units)

A

B

C

D

E

F


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6. What is an example of two rectangles having the same area, but different dimensions?

a. What are the perimeters of these rectangles?

b. If the areas are equal, are the perimeters always equal?

7. You can determine the perimeter of a rectangle without drawing it if you know the

rectangle’s length and width. Explain how you can do this. Use the table in Question 5

to help you.

8. Write a formula that you can use to calculate the

perimeter of any rectangle. Use for the length of

the rectangle, w for the width of the rectangle, and

P for the perimeter.

9. You can determine the area of a rectangle without drawing it if you know the rectangle’s

length and width. Explain how you can do this. Use the table in Question 5 to help you.

The opposite sides of a rectangle

are always the same length.

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10. Write a formula that you can use to calculate the area of any rectangle. Use for

the length of the rectangle, w for the width of the rectangle, and A for the area of

the rectangle.

11. Can you determine the area of a rectangle if its perimeter is known?

Explain your reasoning.

12. Can you determine the perimeter of a rectangle if its area is known?



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13. For each rectangle, either the length, width, or area is unknown. First, calculate the

value of the unknown measure. Then, calculate the perimeter.

a.

21 feet

15 feet

b.

8 millimeters

Area:48 squaremillimeters

c.

Area: 15.75 square inches 3.5 inches

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14. Calculate the perimeter and area of a rectangle that is 11 meters long and

5 meters wide.

a. Double the length and width of the rectangle. Calculate the perimeter of the

new rectangle.

b. What effect does doubling the length and width have on the perimeter?

c. Do you think that doubling the length and width will have the same effect on the

area? Explain your reasoning.

15. Calculate the area of the rectangle that had its dimensions doubled.

16. What effect does doubling the length and width have on the area?


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Problem 2 A Square Rug

Tyson has also received several requests to create square-shaped rugs.

1. One square rug is seven feet long and seven feet wide. Draw a model of this rug on

the grid shown. Each square on the grid represents a square that is one foot long and

one foot wide.

2. What is the area of this rug? Explain your calculation.

3. What is the perimeter of this rug? Explain your calculation.

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4. Six different squares are drawn on the grid. The letters A through F name each square.

Each square on the grid represents a square that is one foot long and one foot wide.

A

D

E

F

B

C

5. Complete the table to show the length, width, area, and perimeter of each square.

Square Length (units)Width (units)

Perimeter (units)

Area (square units)

A

B

C

D

E

F


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6. You can determine the perimeter of a square without drawing the square if you know

the length of one side of the square. Explain how you can do this. Use the table in

Question 5 to help you.

7. Write a formula that you can use to calculate the perimeter of any square.

Use s for the side length of the square and P for the perimeter.

8. You can determine the area of a square without drawing it if you know the length of

one side of the square. Explain how you can do this. Use the table in Question 5 to

help you.

9. Write a formula that you can use to calculate the area of any square. Use s for the

length of a side of the square and A for the area of the square.

10. Calculate the value of the unknown side length, area, and perimeter in each square shown.

a.

5 centimeters

b.

Area: 169 square feet

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11. Calculate the perimeter and area of a square that has a side length equal to 9 inches.

a. Double the side length of the square. Calculate the perimeter of the new square.

b. What effect does doubling the side length of a

square have on the perimeter?

c. Do you think that doubling the side length of a

square will have the same effect on the area?


12. Calculate the area of a square that has a side length equal to

10 meters.

a. Double the length of the side of the square. Calculate the area of

the new square.

b. What effect does doubling the length of a side of a square have on the area?

Is your reasoning about the

effect of changing side lengths of a square, the same or different than your reasoning when you

considered side length changes of a rectangle?


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Problem 3 A Brand New Floor

A carpeting company has been hired to install flooring on the first floor of a home.

A diagram of this first floor is shown.

DiningroomKitchen

Livingroom

Enclosed porch

10 feet

10 feet

10 feet

5 feet

35 feet

8 feet

14 feet

12 feet

b

a

1. Calculate the unknown lengths a and b.

2. The homeowners would like to install indoor/outdoor carpeting on the

enclosed porch. How many square feet of indoor/outdoor carpeting

will be needed?

Use what you do know from the diagram to figure out what you don't know.

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3. The homeowners would like to install wood flooring in the dining room. How many

square feet of wood flooring will be needed?

4. The homeowners would like to install tile in the kitchen. How many square feet of tile

will be needed?

5. The homeowners would like to install loop carpeting in the living room. How many

square feet of loop carpeting will be needed?

6. What is the total area of the first floor? Explain your reasoning.


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7. An employee from the flooring company must now calculate the total cost of the

materials used to complete this job.

a. The carpeting for the porch costs $1.20 per square foot. Calculate the cost of

carpeting the enclosed porch.

b. The loop carpeting costs $0.84 per square foot. Calculate the cost of carpeting

the living room.

c. The wood flooring costs $4.50 per square foot. Calculate the cost of installing the

wood flooring in the dining room.

d. The tiling costs $4.25 per square foot. Calculate the cost of tiling the kitchen.

e. Calculate the total cost of the materials needed for the job.

f. Do you think the total cost is accurate? Why or why not?

g. After the homeowners saw the total cost of the flooring for the job, they decided that

the wood flooring was too expensive and decided to use the same loop carpeting in

the dining room as in the living room. Calculate the total cost of the flooring for the first

floor if the homeowners decide to buy the loop carpeting for the dining room.


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13.3 Area of Parallelograms and Triangles • 847

On you mark . . . get set . . . sail? Sailboat racing isn’t quite started like that, but there is definitely a starting and finish line in this sport. Generally, each

sailboat has a crew that mans various sails. In fact, while you may think that

sailboats can only go in the direction of the wind, that notion is not quite true!

There are ways that sailboats can sail against the wind. In fact, there is almost an

entire science to sailing. How do you think that a sailboat can sail against the

wind? Do you think those same principles can be used for toy sailboats?

Key Terms altitude of a parallelogram

height of a parallelogram

altitude of a triangle

height of a triangle


Calculate the area of parallelograms and triangles.

Write a formula for the area of a parallelogram and a triangle.


BoundaryLinesArea of Parallelograms and Triangles


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Problem 1 A Parallelogram Rug

Tyson has a special request from a client. The client would like a rug in the shape of a

non-rectangular parallelogram. A model of the rug is shown on the grid. Each square

on the grid represents a square that is one foot long and one foot wide.

1. Explain how you can create a rectangle from the figure shown so that the two figures

have the same area. Then, check your answer by demonstrating your method on a

separate sheet of grid paper. Draw your rectangle on top of the figure on the grid.

2. What is the area of the rectangle from Question 1? Explain your calculation.

3. What is the area of the rug? Explain your calculation.

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4. Tyson’s client requests another rug. A model of the new rug is shown on the grid.

Calculate the area of the rug. Explain your reasoning. Each square on the grid

represents a square that is one foot long and one foot wide.

Any side of a parallelogram is a base. Parallelogram EFGH shown is

drawn in different orientations. Each square on the grid represents a

square that is one foot long and one foot wide.

base

base

F

E

E

H G

F

H

G

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An altitudeofaparallelogram is a line segment drawn from a vertex, perpendicular to

the line containing the opposite side. A heightofaparallelogram is the perpendicular

distance from any point on one side to the line containing the opposite side.

base

altitude

altitude

base

F

E

E

H G

F

H

G

5. For each parallelogram, draw a segment that represents a height. Label the height

with its measure and label the base with its measure. Each square on the grid

represents a square that is one foot long and one foot wide.

6. Write a formula for the area of a parallelogram. Use b for the base of the

parallelogram, h for the height, and A for the area.

So, the height is just the length of the altitude.

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7. For each parallelogram, the length of a base, the height, or the area is unknown.

Calculate the value of each unknown measure.

a.

10 feet

9.5 feet

b.

15 meters

Area: 60 square meters

c.

3.5 inches

Area: 28 square inches

Remember, multiplication and

division are inverse operations.


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8. Tyson charges $20 per square foot of rug for a basic design. A client orders one basic

rectangular-shaped rug that is 6 feet long and 4 feet wide and one basic rug shaped

like a parallelogram that is not a rectangle with a base that is 8 feet long and a height

that is 3 feet. What is the total cost for the rugs? Explain your reasoning.

Problem 2 The Race Course

One of the typical shapes of a sailboat race course is triangular.

The course path is identified by buoys called marks. When the

course is triangular-shaped, the marks are located at the

vertices of the triangle.

A sample course with the marks numbered is shown. Each square on the grid represents a

square that is one tenth of a kilometer long and one tenth of a kilometer wide.

12

3

Wind

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1. How many grid squares in a row create an area that is one kilometer long and one

tenth of a kilometer wide?

2. How many grid squares are in an area that is one kilometer long and one kilometer wide?

3. Estimate the area enclosed by the course. Justify your estimate.

4. Is your area from Question 3 exact? Explain your reasoning.

5. Use two sides of the triangle to draw a parallelogram on the grid.

6. Calculate the area of the parallelogram you drew.

7. Can you calculate the exact area of the triangle by using the area of the

parallelogram? Why or why not?


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8. Calculate the exact area enclosed by the triangular course.

9. How does the exact area enclosed by the triangular-shaped course compare to

the estimate?

10. How does the area of the parallelogram relate to the

area of the triangle?

11. Consider the race course shown on the grid. Each square on the

grid represents a square that is one tenth of a kilometer long and

one tenth of a kilometer wide. Calculate the area enclosed by

the course.

1 2

3

But why are the two

triangles that form a parallelogram

congruent?

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12. What information about the triangle did you need to calculate the area in Question 11?

Any side of a triangle is a base.

Triangle KYM is the same triangle drawn in three different orientations.

baseY

K

M baseK Y

M

baseM K

Y

An altitudeofatriangle is a line segment drawn from a vertex perpendicular to a line

containing the opposite side. A heightofatriangle is the perpendicular distance from a

vertex to the line containing the base.

altitudealtitudealtitude

baseY

K

M baseK Y

M

baseM K

Y

13. Write a formula that you can use to calculate the area of any triangle. Use b for the

length of the base, h for the height, and A for the area of the triangle.


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14. Determine the base and height of triangles KYM, MYK, KMY. Then calculate the area

of each triangle.

baseY

K

M baseK Y

M

baseM K

Y

15. Describe what happens to the height of a triangle as the length of the base changes

when the area remains the same.

16. For each triangle, the length of the base, the height, or the area is unknown. Calculate

the value of each unknown measure.

a.

24 meters


b.

6 yards

8 yards

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c.

6 inches

Area: 42 square inches

17. The original race course is shown, but now the lengths of the legs of the race are

given. If a boat must complete the course once, how long is the race?

12

3

Wind

1.6 kilometers

1.6 kilometers 1.6 kilometers

18. What geometric name is given to this measurement?

19. In sailboat races, it is common for a boat to have to go around a course more than

once or revisit a leg of the course more than once. Suppose that to complete the

race, a boat must sail to the marks in the following order: 1, 2, 3, 1, 3, 1, 2, 3, 1, 3.

How long is this race?


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20. If a boat is competing in a race, do you think that the boat will travel more than or less

than the race length you calculated in Question 19.

Problem 3 Boundary Lines

1. Each square on the grid formed by connecting four closely positioned dots represents

a square that is 1 unit long and 1 unit wide. Line KN is drawn parallel to line PR.

K

P R

M N

a. Calculate the area of triangle KPR.

b. Calculate the area of triangle MPR.

c. Calculate the area of triangle NPR.

d. Compare the areas of triangles KPR, MPR, and NPR.

e. Compare the bases of triangles KPR, MPR, and NPR.

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f. Compare the heights of triangles KPR, MPR, and NPR.

g. What conclusion can be made about triangles that share the same base, or have

bases of equal measure, and also have equal heights?

2. Use the conclusion you made in Question 1, part (g) to solve this problem.

A sister and brother inherit equal amounts of property; however, the boundary line

separating their land is not straight. Your job is to draw a new boundary line that is

straight and keeps the property division equal. Explain how you solved the problem.

A Upper boundary line

Lower boundary line

Sister’sLand

Brother’sLand

B

Also think about the area

relationships you already established about parallelograms and

triangles.


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Problem 4 Composite Figures

1. The figure shown is composed of a rectangle and triangles.

60 ft

20 ft

R

CE

TA

a. Describe a strategy that can be used to compute the area of the shaded region.

b Calculate the area of rectangle RECT.

c. Calculate the area of triangle AEC.

It really helps to think about a few

strategies before jumping right in!

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d. Calculate the area of the shaded region.

2. The figure shown is composed of rectangles and triangles.

R

AN

T C

E

5"1" 1"

1.5"

XH

GO

a. Describe two different strategies that can be used to compute the area of the

entire shaded region.

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b. Use one of your strategies to calculate the area of the shaded region.


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13.4 Area of Trapezoids • 863

In most classrooms, a projection screen is hung above the blackboard. Generally, the screen is located higher than where the projector sits. To view images on the

screen, the projector must be tilted upward. This tilting can cause “keystoning,”

which is a distortion of the image. A normal image and possible distorted image

are shown.

Four things to remember when using the Internet:

• Never share your personal information, such as addresses, phone numbers, or photographs, with online friends. • Never agree to meet someone face-to-face you met online. • Never respond to messages or bulletin boards that make you feel uncomfortable. • People you meet online may not be who they say they are.

Key Terms bases of a trapezoid

legs of a trapezoid

altitude of a trapezoid

height of a trapezoid


Calculate the area of trapezoids.

Write a formula for the area of a trapezoid.


TheKeystoneEffectArea of Trapezoids

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Problem 1 How Distorted?

1. Describe how the normal image from the lesson opener has been distorted.

2. Describe the shapes formed by the normal image and the distorted image.

3. Which image do you think has a larger area? Explain your reasoning.

4. The normal image and the distorted image are shown on the grid. Each square on the

grid represents a square that is four inches long and four inches wide. Calculate the

area of each image and write it in the center of the image.

5. How do the areas of the images compare?

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6. Is your area of the distorted image exact? Explain your reasoning.

7. Consider the distorted image in Question 4. How can you use the area formulas you

already know to calculate the exact area of this image?

8. Calculate the exact area of the distorted image.


Pay attention to the scale on

the grid!


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9. How do the exact areas of the normal image and the distorted image compare?

10. Consider the distorted image again. Suppose that you make an exact copy of this

image, flip it vertically, and move it next to the image as shown.

a. What is the geometric figure that is formed from these images?

b. Use a formula to calculate the area of the parallelogram. Then, use that area to

calculate the area of the distorted image.

You did something just

like this when you related the areas of

parallelograms and triangles.

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Trapezoid TRAP is the same trapezoid drawn in different orientations.

P base

base

base base base base

base

baseA T

T

T

P A

R

T

PA

R

R P A

R

11. Was it easier to calculate the area of the distorted image by using your method in

Question 7 or by using the method in Question 10? Explain your reasoning.

The distorted image is a trapezoid. The parallel sides of the trapezoid are called the bases

ofthetrapezoid. Non-parallel sides are thelegsofthetrapezoid.


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An altitudeofatrapezoid is a line segment drawn from a vertex perpendicular to a line

containing the opposite side. A heightofatrapezoid is the perpendicular distance from a

vertex to the line containing the base.

P base

altitude

altitude

altitude

altitude

base base

baseA T

T

T

P A

R

T

PA

R

R P A

R

12. Consider the trapezoid shown. Suppose that you make

an exact copy of this trapezoid, flip it vertically, and

move it next to the trapezoid as shown. Label the bases

of the trapezoid on the right.

h

b1

b2

a. Write a formula for the area of the entire figure.

b. Write a formula for the area of one of the trapezoids.


Can you prove to me that this figure is a parallelogram?

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13. For each trapezoid, either a height, the length of one base, or the area is unknown.

Determine the value of each unknown measure.

a. 22 millimeters

6 millimeters

8 millimeters

b. 3 feet

7 feet

Area: 25 square feet

c.

6 meters

9 meters



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14. The projector in Problem 1 was tilted differently to create the distorted image

shown. Each square on the grid represents a square that is four inches long and

four inches wide.

Normal Image Distorted

Image

a. What is the area of the distorted image?

b. How does the area of the distorted image compare to the area of the

normal image?


1. The figure shown is composed of a rectangle and four congruent trapezoids.

30 cm

22 cm 80 cm

T

C

R

E

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a. Describe a strategy that can be used to compute the area of the shaded region.

b. Calculate the area of rectangle RECT.

c. Calculate the area of a trapezoid.

d. Calculate the area of the shaded region.


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2. The figure is composed of a rectangle and a regular hexagon.

The length of each side of the hexagon is 2 centimeters.

4

3.5

2 cm

a. Describe two strategies that can be used to compute the area of the shaded

region.

b. Calculate the area of the shaded region.

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Talk the Talk

Write the area formula for each figure.

1.

<

w

2.

s

s

Area of a Rectangle Formula: Area of a Square Formula:

3.

b

h

4.

b

h

Area of a Parallelogram Formula: Area of a Triangle Formula:

5.

h

b1

b2

Area of a Trapezoid Formula:



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13.5 Area of Rhombi and Kites • 875

GoFlyaKiteArea of Rhombi and Kites


Calculate the area of rhombi and kites.

Use formulas to compute the area of rhombi and kites.


What hobby’s history is older than our current calendar? If you said kites, you’d be correct. It is thought that kites were created around 2800 years ago in

China. Early kites were made of silk and had tails. From China, the kite traveled to

India where it evolved to a different type of kite called a fighter kite, or patang.

Fighter kiting is still in existence today. Competition can be tough as each kite

flyer tries to ground his or her opponent’s kite through various maneuvers. And

let’s not forget one of the most famous kites that may or may not have been

flown. In 1750, Benjamin Franklin proposed flying a kite during a lightning storm to

prove that electricity existed in lightning. Have you ever flown a kite? How do you

think Benjamin Franklin could have used a kite to prove electricity exists in

lightning?

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Problem 1 Area of a Rhombus

Recall that a rhombus is a quadrilateral with all sides congruent.

Mr. Gram asked his math students to sketch a parallelogram. These are two of his

students’ sketches.

Molly’s Sketch

A B

CD

4 cm

4 cm

2 cm2 cm

James’s Sketch

E F

GH

3 cm

3 cm

3 cm3 cm

1. How are the two sketches similar?

2. How are the two sketches different?

3. Mr. Gram told his class that both sketches are examples of parallelograms, but

James’s sketch is an example of a special parallelogram. Why would James’s sketch

be considered a special parallelogram?

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4. Mr. Gram wrote two statements on the board.

• All parallelograms are rhombi.

• All rhombi are parallelograms.

Are both of these statements true? Explain your reasoning.

5. Sketch a parallelogram that is not a rhombus.

6. Sketch a rhombus that is not a parallelogram.

7. Molly concluded that all squares must be rhombi and parallelograms. Is she correct?


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8. Mr. Gram told Molly that he is thinking of a quadrilateral that is either a square or a

rhombus that is not a square. He wants Molly to guess which quadrilateral he is

thinking of, and he allows her to ask one question about the quadrilateral. Which

question should she ask?

9. Since all rhombi are also parallelograms, what formula can be used to calculate

the area of a rhombus? Use b for the length of the base, h for the height, and A for

the area.

Problem 2 Area of a Kite

Recall that a kite is a quadrilateral with two pairs of consecutive congruent sides with

opposite sides that are not congruent.

1.5 cm

4 cm

4 cm

C

B

A

D

1.5 cm

Kite ABCD

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1. Mr. Gram sketched the kite shown. He asked his

students to add a line segment such that it would

divide the kite into two familiar figures.

1.5 cm

4 cm

4 cm

C

B

A

D

1.5 cm

Molly’s kite

1.5 cm

4 cm

4 cm

C

B

A

D

1.5 cm

James’s kite

If Mr. Gram asked you to determine the area of the kite, would you rather use Molly’s

kite or James’s kite? Explain.

2. Describe a strategy that can be used to compute the area of the kite.

3. If you are using Molly’s kite to calculate the area of the kite, what additional

information would you need?

So, it can't just look like a

kite. In mathematics, kite has a very specific

definition.”

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4. If you are using James’s kite to calculate the area of the kite, what additional

information would you need?

5. How would the area of Molly’s kite compare to the area of James’s kite?

6. Given:

AC = 5 cm BD = 2.5 cm

AE = 1.1 cm BE = 1.25 cm

CE = 3.9 cm DE = 1.25 cm 1.5 cm

4 cm

4 cm

CE

B

A

D

1.5 cm

Calculate the area of kite ABCD using both Molly’s and James’s strategies.

Did your calculations verify

your predictions about the area of Molly's kite compared to the area

of James's kite?

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1. The figure shown is composed of a rectangle and a kite.

C

B

A

D

E

Describe a strategy that can be used to compute the area of the shaded region.

2. How do you think the areas of the triangles in the kite compare to the areas of the

triangles in the shaded region?

3. How do you think the area of the kite compares to the area of the shaded region?


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4. Given:

AC = 10 m

BE = 4 m

Calculate the area of the kite.

5. Calculate the area of the rectangle.

6. Calculate the area of the shaded region.


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StreetSignsArea of Regular Polygons

13.6 Area of Regular Polygons • 883

Learning GoalsIn this lesson you will:

Calculate the area of regular polygons.

Write a formula for the area of a regular polygon.


Have you ever noticed that every stop sign looks exactly the same, every yield sign looks exactly the same, and so on? This is because the Federal Highway

Administration has standards that indicate the exact sizes and colors of roadway

signs. Most of the sign shapes are polygons.

Key Terms congruent polygons

apothem

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Problem 1 How Big is that Sign?

The specifications for the smallest possible yield sign are shown.

YIELD

30 in.60o 60o

60o

30 in.30 in.

1. What is special about the triangle that forms the yield sign?

The specifications for a “Do Not Enter” sign are shown.

DO NOT

ENTER

30 in.

90o 90o

90o 90o

30 in.30 in.

30 in.

2. What is special about the quadrilateral that forms the “Do Not Enter” sign?

The specifications for a stop sign are shown.

STOP

135o

135o 135o

135o

135o 135o

135o 135o

12.4 in.

12.4 in.

12.4 in.

12.4 in.

12.4 in.

12.4 in.

12.4 in.

12.4 in.

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3. What is special about the octagon that forms the stop sign?

The polygons in Questions 1 through 3 are special polygons called regular polygons. Two

other possible sizes for a yield sign are shown.

36 in.

36 in.36 in.

60o 60o

60o

48 in.60o 60o

48 in.48 in.

60o

4. Are these signs regular polygons? What can you conclude about all regular triangles?

5. The yield sign from Question 1 is shown with its approximate height. Calculate the

approximate area of the yield sign.

30 in.

26 in.

YIELD

6. Calculate the approximate area of the “Do Not Enter” sign from Question 2.

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When two polygons are exactly the same size and exactly the same shape, the polygons

are said to be congruentpolygons.

7. To calculate the area of the stop sign from Question 3, you can use the fact that a

regular polygon can be divided into triangles that are all exactly the same size and

same shape. The bases of the triangles are the sides of the polygon as shown. In this

case, the height of each triangle is approximately 15 inches. Calculate the area of the

stop sign. Round your answer to the nearest tenth if necessary. Explain your reasoning.

12.4 in.15 in.

The height of the triangle in the stop sign in Question 7 is

the apothem of the octagon. The apothem of a regular

polygon is the perpendicular distance from the center of the

regular polygon to a side of the regular polygon.

8. Draw a segment that represents an apothem on

each regular polygon shown. The center of the

polygon is marked by a point.

Many dictionaries say the correct

pronunciation of apothem is AP-uh-thum.

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9. The hexagon shown is a regular hexagon. Calculate the area of the hexagon. Explain

your reasoning.

40 cm

34.6 cm

10. The heptagon shown is a regular heptagon. Calculate the area of the heptagon.


8 m

8.3 m

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11. Explain how you can calculate the area of a regular polygon if you know the length of

the apothem and the length of each side.

12. Write a formula for the area of a regular polygon with n sides. Use a for the length of

the apothem and for the length of one side of the polygon.

Problem 2 Perimeter and Apothems

1. Lily claims the formula for determining the area of a regular polygon is A 5 ( 1 __ 2 a ) n, where is the length of a side, a is the apothem, and n is the number of sides.

Molly claims the formula for determining the area of a regular polygon is A 5 1 __ 2 Pa,

where P is the perimeter of the polygon and a is the apothem.

Who is correct? Explain your reasoning.

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2. Emma thinks the definition for a regular polygon is too long and it should be

shortened. She believes that if a polygon has all sides equal in length, then all angles

will always be equal in measure.

a. What are two examples Emma could use to justify her conclusion?

b. Is Emma correct? Justify your conclusion.

3. Jath also thinks the definition for a regular polygon is too long. He states that if a

polygon has all angles of equal measure, then all sides will always be equal in length.

a. What are two examples Jath could use to justify his conclusion?

b. Is Jath correct? Justify your conclusion.

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Problem 3 Calculate the Area

1. The length of one side of a regular nonagon is 24 feet, and the length of the apothem

is approximately 33 feet. Calculate the area of the regular nonagon.

2. The side length of the largest possible stop sign is 20 inches, and the length of the

apothem is approximately 24.1 inches.

a. What is the area of the largest possible stop sign?

b. The side length of the smallest possible stop sign is 9.9 inches, and the length of

the apothem is approximately 12 inches. What is the area of the smallest possible

stop sign?

c. How many times larger is the area of the largest possible stop sign than the area

of the smallest possible stop sign? Explain your reasoning.

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Problem 4 How Big Is the Pentagon?

The United States Defense Department is located in a building called the Pentagon in

Arlington, Virginia. This regular pentagonal-shaped building has 17.5 miles of corridors.

Each side of the building is approximately 921 feet (307 yards) long, and the apothem of

the pentagon is approximately 633.8 feet (211.27 yards).

1. Determine the approximate area of the ground level of the Pentagon in square yards.

2. Determine the area of a football field.

3. How does the area of a football field compare to the area on

the ground level of the Pentagon?

A football field is 50 yards

wide and 100 yards long.

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The figure shown is composed of two regular hexagons.

12 mm

20 mm

10 mm

6 mm

1. Describe a strategy that can be used to compute the area of the shaded region.

2. Calculate the area of the shaded region.


Think back to all the composite figures you

have worked with in this chapter. Did you use similar strategies to calculate the area of

the shaded region in each?

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Sketching and Drawing Polygons

When you sketch a geometric figure, the figure is created

without the use of tools. When you draw a geometric

figure, the figure is created with the use of tools such

as a ruler, a straightedge, a compass, or a protractor.

A protractor can be used to approximate the

measure of an angle. A compass is a tool used to

create arcs and circles. A straightedge is a ruler with

no numbers.

Example

A sketch and a drawing of a parallelogram are shown.

sketch drawing

Think the size or shape of your brain

matters? It doesn't. In fact your brain keeps growing and changing throughout your life

every time you learn something new!

Key Termsprotractor (13.1)

compass (13.1)

straightedge (13.1)

sketch (13.1)

draw (13.1)

construct (13.1)

triangle (13.1)

equilateral triangle (13.1)

isosceles triangle (13.1)

scalene triangle (13.1)

equiangular triangle (13.1)

acute triangle (13.1)

right triangle (13.1)

obtuse triangle (13.1)

quadrilateral (13.1)

opposite sides (13.1)

consecutive sides (13.1)

square (13.1)

rectangle (13.1)

rhombus (13.1)

parallelogram (13.1)

kite (13.1)

trapezoid (13.1)

isosceles trapezoid (13.1)

polygon (13.1)

regular polygon (13.1)

irregular polygon (13.1)

pentagon (13.1)

hexagon (13.1)

heptagon (13.1)

octagon (13.1)

nonagon (13.1)

decagon (13.1)

altitude of a parallelogram (13.3)

height of a parallelogram (13.3)

altitude of a triangle (13.3)

height of a triangle (13.3)

bases of a trapezoid (13.4)

legs of a trapezoid (13.4)

altitude of a trapezoid (13.4)

height of a trapezoid (13.4)

congruent polygons (13.6)

apothem (13.6)

Chapter 13 Summary

Chapter 13 Summary • 893


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Naming Polygons and their Sides and Angles

A polygon is a closed figure that is formed by joining three or more line segments at their

endpoints. A regular polygon is a polygon with all sides congruent and all angles

congruent. An irregular polygon is a polygon that is not regular. Polygons are named using

capital letters representing the vertices, listed in clockwise or counterclockwise order.

Example

Right triangle XYZ is shown.

X

ZY

Triangle XYZ can also be named nYZX, nZXY, nXZY, nZYX, or nYXZ. The sides of nXYZ

are XY, YZ, and ZX. The angles of nXYZ are ∠X, ∠Y, and ∠Z.

Sorting Polygons into Categories

Polygons are classified by the number of sides they have.

● Triangles have 3 sides.

● Quadrilaterals have 4 sides.

● Pentagons have 5 sides.

● Hexagons have 6 sides.

● Heptagons have 7 sides.

● Octagons have 8 sides.

● Nonagons have 9 sides.

● Decagons have 10 sides.

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Some quadrilaterals have special names because of their specific qualities. A square is a

quadrilateral with all sides congruent and all angles congruent. A rectangle is a

quadrilateral with opposite sides congruent and all angles congruent. A rhombus is a

quadrilateral with all sides congruent. A parallelogram is a quadrilateral with both pairs of

opposite sides parallel. A kite is a quadrilateral with two pairs of consecutive congruent

sides with opposite sides that are not congruent. A trapezoid is a quadrilateral with exactly

one pair of parallel sides. An isosceles trapezoid is a trapezoid whose non-parallel sides

are congruent.

Example

These regular polygons have been sorted into two groups. One group contains regular

polygons whose opposite sides are parallel and the other group contains regular polygons

without any opposite sides that are parallel.

Regular polygons with opposite sides parallel. Regular polygons without opposite sides parallel.

Square Regular Hexagon Regular Triangle Regular Pentagon

Regular Octagon Regular Decagon Regular Heptagon Regular Nonagon

Notice that all of the regular polygons with opposite sides parallel have an even number of

sides. Similarly, all of the regular polygons without opposite sides parallel have an odd

number of sides.


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Calculating the Perimeter and Area of a Rectangle

To calculate the perimeter, P, of a rectangle, use the formula P 5 2,1 2w, where ,

represents the length of the rectangle and w represents the width of the rectangle. To

calculate the area, A, of a rectangle, use the formula A 5 ,w.

Example

5 ft

16 ft

P 5 2, 1 2w 5 2(16) 1 2(5) 5 32 1 10 5 42 feet

A 5 ,w 5 16(5) 5 80 square feet

The perimeter of the rectangle is 42 feet, and the area of the rectangle is 80 square feet.

Calculating the Perimeter and Area of a Square

To calculate the perimeter, P, of a square, use the formula P 5 4s, where s is the side

length of the square. To calculate the area, A, of a square, use the formula

A 5 s 3 s or A 5 s2.

Example

12 m

12 m

P 5 4s 5 4(12) 5 48 meters

A 5 s · s 5 s2 5 122 5 144 square meters

The perimeter of the square is 48 meters, and the area of the square is 144 square meters.

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Calculating the Area of Composite Figures

Composite figures are figures that are composed of multiple geometric shapes. To

determine the area of a composite figure, it is usually best to separate the figure into its

composite shapes, calculate the area of each shape, and then add these areas together to

determine the total area.

Example

The figure shown is a composite figure consisting of one square and two congruent rectangles.

2 yards

6 yards

6 yards

10 yards

2 yards

The length of each side of the square is 6 yards. The area of the square is A 5 s · s 5 s2 5

62 = 36 square yards. Each rectangle has a length of 4 yards and a width of 2 yards. The

area of one rectangle is A 5 ,w 5 4(2) 5 8 square yards. The total area of the composite

figure is 36 1 2(8) 5 36 1 16 5 52 square yards.

Calculating the Area of a Parallelogram

To calculate the area, A, of a parallelogram, use the formula A 5 bh, where b represents the

length of the base of the parallelogram and h represents the height of the parallelogram.

Example

22 m

10.5 m

A 5 bh

5 22(10.5)

5 231 square meters

The area of the parallelogram is 231 square meters.


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Calculating the Area of a Triangle

To calculate the area, A, of a triangle, use the formula A 5 1 __ 2

bh, where b represents the

length of the base of the triangle and h represents the height of the triangle.

Example

11 cm

7 cm

A 5 1 __ 2

bh

5 1 __ 2

(11)(7)

5 38.5 square centimeters

The area of the triangle is 38.5 square centimeters.

Calculating the Area of a Trapezoid

To calculate the area, A, of a trapezoid, use the formula A 5 1 __ 2

(b1 1 b2)h, where b1 and b2

represent the lengths of the two bases and h represents the height of the trapezoid.

Example

65 yd

35 yd

30 yd

A 5 1 __ 2

(b1 1 b2)h

5 1 __ 2 (35 1 65)(30)

5 1500 square yards

The area of the trapezoid is 1500 square yards.

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Calculating the Area of a Rhombus

All rhombi are parallelograms. Therefore, to calculate the area of a rhombus, use the area

formula for a parallelogram.

Example

Each side of the rhombus shown has a length of 10 inches.

10 in

8.5 in

A 5 bh

5 10(8.5)

5 85 square inches

The area of the rhombus is 85 square inches.


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Calculating the Area of a Kite

A kite can be thought of as a composite figure consisting of two triangles. To calculate the

area of a kite, calculate the area of each triangular region and then add the two areas

together.

Example

12 ft

7 ft

The kite is composed of two triangles, each with a base length of 12 feet and a height of

3.5 feet.

The area of one triangle is:

A 5 1 __ 2

bh

5 1 __ 2

(12)(3.5)

5 21 square feet

The area of the kite is 2(21) or 42 square feet.

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Calculating the Area of a Regular Polygon

The apothem of a regular polygon is the perpendicular distance from the center of the

polygon to a side of the polygon. To calculate the area, A, of a regular polygon, use the

formula A 5 1 __ 2

Pa, where P represents the perimeter of the polygon and a represents the

length of the apothem.

Example

13.8 m

20 m

The perimeter of the pentagon is 5(20) or 100 meters.

A 5 1 __ 2

Pa

5 1 __ 2

(100)(13.8)

5 690 square meters

The area of the pentagon is 690 square meters.

Area · 2017. 4. 4. · A quadrilateral is a closed four-sided geometric figure. Opposite sides are...

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