Reteach 9-2 Developing Formulas for Circles and Regular...

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Name Date Class

LESSON ReteachDeveloping Formulas for Circles and Regular Polygons

Circumference and Area of Circles

A circle with diameter d and radius r has circumference C � �d or C � 2�r.

A circle with radius r has area A � �r 2.

Find the circumference of circle S in which A � 81� cm2.

Step 1 Use the given area to solve for r.

cm

A � �r 2 Area of a circle

81� cm2 � �r 2 Substitute 81� for A.

81 cm2 � r 2 Divide both sides by �.

9 cm � r Take the square root of both sides.

Step 2 Use the value of r to find the circumference.

C � 2�r Circumference of a circle

C � 2�(9 cm) � 18� cm Substitute 9 cm for r and simplify.

Find each measurement.

1. the circumference of circle B 2. the area of circle R in terms of �

6– cm

5 m

C � 6 cm A � 25� m2

3. the area of circle Z in terms of � 4. the circumference of circle T in terms of �

22 ft

10 in.

A � 121� ft 2 C � 20� in.

5. the circumference of circle X in 6. the radius of circle Y in which C � 18� cmwhich A � 49� in2

C � 14� in. r � 9 cm

9-2


Name Date Class

LESSON ReteachDeveloping Formulas for Circles and Regular Polygons continued9-2

Area of Regular Polygons

The area of a regular

polygon with apothem

a and perimeter P

is A � 1 __ 2 aP.

Find the area of a regular hexagon with side length 10 cm.

Step 1 Draw a figure and find the measure of a central angle. Each central

angle measure of a regular n-gon is 360° ____ n .

Step 2 Use the tangent ratio to find the apothem. You could also use the 30°-60°-90° � Thm. in this case.

tan 30° � leg opposite 30° angle

____________________ leg adjacent to 30° angle

Write a tangent ratio.

tan 30° � 5 cm _____ a Substitute the known values.

a � 5 cm ______ tan 30°

Solve for a.

Step 3 Use the formula to find the area.

A � 1 __ 2 aP

A � 1 __ 2 � 5 ______

tan 30� � 60 a � 5 ______

tan 30° , P � 6 � 10 or 60 cm

A � 259.8 cm2 Simplify.

Find the area of each regular polygon. Round to the nearest tenth.

7. 12 cm 8.

4 in.

A � 695.3 cm2 A � 58.1 in2

9. a regular hexagon with an apothem of 3 m 10. a regular decagon with a perimeter of 70 ft

A � 31.2 m2 A � 377.0 ft2

The apothem is the distance from the center to a side.

The center is equidistant from the vertices.

A central angle has its vertex at the center. This central angle measure is

360� ____ n � 60�.



LESSON

9-2Practice ADeveloping Formulas for Circles and Regular Polygons

In Exercises 1–3, fill in the blanks to complete each formula.

1. The area of a regular polygon with apothem a and perimeter P is A �

1__2

aP .

2. A circle with diameter d has circumference C � �d .

3. A circle with radius r has area A � �r 2 .

Use the area and circumference formulas for circles to find each measurement. Give your answers in terms of �.

4.

5 ft

5.

20 in.

the area of �A the area of �Q

A � 25� ft2 A � 100� in2

6.

18 cm

7.13 mi

the circumference of �W the circumference of �N

C � 18� cm C � 26� mi

8. the radius of �I in which A � 144� meters2 12 m

9. the diameter of �L in which C � 2� kilometers 2 km

10. the area of �P in which C � 32� yards 256� yd2

11. Emile is shopping for a new bicycle. He sees that a trick bike has 20-inch-diameter wheels, a mountain bike has 26-inch-diameter wheels, and a racing bike has 27-inch-diameter wheels. Find the area of each wheel. Round to the nearest tenth.

314.2 in2; 530.9 in2; 572.6 in2

Use the formula for the area of a regular polygon to find each measurement.

12.

2.3 ft

8 ft

13.

18.1 mm15 mm

the area of the regular triangle the area of the regular octagon

A � 27.6 ft2 A � 1086 mm2

14. the side length of a regular nonagon in which A � 99 in2 and a � 5.5 in. s � 4 in.


LESSON

9-2Practice BDeveloping Formulas for Circles and Regular Polygons

Find each measurement. Give your answers in terms of �.

1.25 m

2.

4 in.

the area of �V the area of �H

A � 625� m2 A � 4a 2� in2

3.

( ) yd

4.

1200 mi

the circumference of �M the circumference of �R

C � (2x � 2y)� yd C � 1200� mi

5. the radius of �D in which C � 2�2 cm r � � cm

6. the diameter of �K in which A � (x 2� 2x � 1)� km2 d � (2x � 2) km

Stella wants to cover a tabletop with nickels, dimes, or quarters. She decides to find which coin would cost the least to use.

7. Stella measures the diameters of a nickel, a dime, and a quarter. They are 21.2 mm, 17.8 mm, and 24.5 mm. Find the areas of the nickel, the dime, and the quarter. Round to the nearest tenth.

353.0 mm2; 248.8 mm2; 471.4 mm2

8. Divide each coin’s value in cents by the coin’s area. Round to the nearest hundredth.

0.01 cent/mm2; 0.04 cent/mm2; 0.05 cent/mm2

9. Tell which coin has the least value per unit of area. the nickel

10. Tell about how many nickels would cover a square tabletop that measures 1 square meter. Then find the cost of the coins.

2833 nickels; $141.65Find the area of each regular polygon. Round to the nearest tenth.

11.

18 in.

12.

6 m

A � 1122.4 in2 A � 85.6 m2


LESSON

9-2Practice CDeveloping Formulas for Circles and Regular Polygons

Write a paragraph proof.

1. Given: A is the center of a regular polygon._AD is an apothem of the regular polygon.

Prove:_AD bisects �BAC and

_BC.

Possible answer: By the definition of the center of a regular polygon,_AC �

_AB, so �ABC is isosceles. By the Isosceles Triangle Theorem,

�ACB � �ABC. By the definition of apothem, �ADC and �ADB are right angles so �ADC � �ADB by the Right Angle Congruence Theorem. _AD �

_AD by the reflexive property, so �ADC � �ADB by AAS. By CPCTC,

�CAD � �BAD and _CD �

_DB. Thus, by the definition of angle bisector

and segment bisector, _AD bisects �BAC and

_BC.

In Exercises 2–5, find the area and the perimeter of a 1 unitregular polygon in which the distance from the center to a vertex is 1 unit. Round to the nearest thousandth.

2. regular octagon 3. regular 14-gon

A � 2.828 units2; P � 6.123 units A � 3.037 units2; P � 6.231 units

4. regular 90-gon 5. regular 1000-gon

A � 3.139 units2; P � 6.282 units A � 3.142 units2; P � 6.283 units

6. Looking at your answers to Exercises 2–5, tell what happens to the ratio of the perimeter to the area as the number of sides increases.

The ratio of the perimeter to the area approaches 2.

7. Provide an explanation for your answer to Exercise 6. Possible answer: As the number of sides in a regular polygon increases, the polygon gets closer to the shape of a circle, the distance from the center to a vertex gets closer to a radius, and the perimeter gets closer to a circumference. In a circle with a radius of 1 unit, the area is � units2 and the circumference is 2� units. So the ratio of the circumference to the area is 2.

Find the number of sides of each regular polygon. (Hint: Use trial and error.)

8. a regular polygon in which A � 90.9 in2 and P � 35 in. 5 or 7

9. a regular polygon in which A � 84.9 cm2 and the side lengths are 14 cm 3

10. a regular polygon in which A � 324.9 m2 and the apothem is 10 m 10


LESSON ReteachDeveloping Formulas for Circles and Regular Polygons

Circumference and Area of Circles

A circle with diameter d and radius r has circumference C � �d or C � 2�r.

A circle with radius r has area A � �r 2.

Find the circumference of circle S in which A � 81� cm2.

Step 1 Use the given area to solve for r.

cm

A � �r 2 Area of a circle

81� cm2� �r 2 Substitute 81� for A.

81 cm2� r 2 Divide both sides by �.

9 cm � r Take the square root of both sides.

Step 2 Use the value of r to find the circumference.

C � 2�r Circumference of a circle

C � 2�(9 cm) � 18� cm Substitute 9 cm for r and simplify.

Find each measurement.

1. the circumference of circle B 2. the area of circle R in terms of �

6– cm5 m

C � 6 cm A � 25� m2

3. the area of circle Z in terms of � 4. the circumference of circle T in terms of �

22 ft10 in.

A � 121� ft 2 C � 20� in.

5. the circumference of circle X in 6. the radius of circle Y in which C � 18� cmwhich A � 49� in2

C � 14� in. r � 9 cm

9-2

001_062_Go07an_CRB_c09.indd 53001_062_Go07an_CRB_c09.indd 53 5/11/06 6:56:51 PM5/11/06 6:56:51 PMProcess BlackProcess Black



Name Date Class

LESSON ReteachDeveloping Formulas for Circles and Regular Polygons continued9-2

Area of Regular Polygons

The area of a regular

polygon with apothem

a and perimeter P

is A � 1__2

aP.

Find the area of a regular hexagon with side length 10 cm.

Step 1 Draw a figure and find the measure of a central angle. Each central

angle measure of a regular n-gon is 360°____n .

�

��

��

Step 2 Use the tangent ratio to find the apothem. You could also use the 30°-60°-90° � Thm. in this case.

tan 30° �leg opposite 30° angle____________________

leg adjacent to 30° angle Write a tangent ratio.

tan 30° � 5 cm_____a Substitute the known values.

a � 5 cm______tan 30°

Solve for a.

Step 3 Use the formula to find the area.

A � 1__2

aP

A � 1__2 � 5______

tan 30� � 60 a � 5______

tan 30° , P � 6 � 10 or 60 cm

A � 259.8 cm2 Simplify.

Find the area of each regular polygon. Round to the nearest tenth.

7. 12 cm 8.

4 in.

A � 695.3 cm2 A � 58.1 in2

9. a regular hexagon with an apothem of 3 m 10. a regular decagon with a perimeter of 70 ft

A � 31.2 m2 A � 377.0 ft2

�

The apothem is the distance from the center to a side.

The center is equidistant from the vertices.

A central angle has its vertex at the center. This central angle measure is360� ____

n � 60�.


Name Date Class

LESSON

9-2ChallengeFinding the Area of the Koch Snowflake

The diagram below shows the first four stages in the construction of the Koch Snowflake. Perhaps you have already worked with this figure. If so, you may have developed some intuition about its area. On this page, you will see how you can use algebraic techniques to actually calculate the area.

�

��

��

��

��

1. What is the area, A0, of the snowflake at stage 0? A0 �

��

3___4

s 2

2. a. What is the area of each triangle added at stage 1?

��

3___4 � s__

3 �2 � �

� 3___

4 � s2

__9

� � ��

3___36

s 2

b. How many triangles are added at stage 1? 3

c. What is the total area, A1, of the snowflake at stage 1? A1 �

��

3___4

s 2

� �

� 3___

12s 2

3. What is the total area, A2, of the

snowflake at stage 2? A2 �

��

3___4

s 2

�

��

3___12

s 2

� �

� 3___

27s 2

4. What is the area, A3, of the

snowflake at stage 3? A3 �

��

3___4

s 2

�

��

3___12

s 2

� �

� 3___

27s 2

�

4��

3____243

s 2

5. After several calculations, it can be shown that a formula for the area of

the snowflake at any stage n, where n is greater than 0, is

An � ��

3___4

s 2� ��

3___12 � 1 � 4__

9� 42

__92 � . . . � 4n � 1

_____9n � 1 � . . . �s2. On a separate sheet of

paper, verify that your answers to Exercises 2– 4 satisfy this formula.

6. In Exercise 5, � 1 � 4__9

� 42__92 �. . . � 4n � 1

_____9n � 1 � . . . � is an infinite geometric series. The ratio

between every set of consecutive terms is 4__9

. A formula for finding a sum like this is

S � a_____1 � r

, where a is the first term and r is the ratio between terms.

Use this formula to find the sum. S � 9__

5

7. Substitute your answer to Exercise 6 into the formula in Exercise 5. Simplify the result. What is the area of the Koch Snowflake?

An � 2��

3____5

s2

Reminder:

The area of an equilateral triangle

with side length s is ��

3___4

s2.

Checkstudents’work.


Name Date Class

LESSON Problem SolvingDeveloping Formulas for Circles and Regular Polygons

1. What is the area of the regular nonagon? Round to the nearest tenth.

8 cm

395.6 cm2

3. When diving and snorkeling, you should leave a “radius of approach,” or a restricted area around certain animals that live in the waters where you are diving. How much greater is the restricted area around a monk seal than the restricted area around a sea turtle? Give your answer in terms of �.

Animal Radius of Approach

sea turtle 20 ft

monk seal 100 ft

9600� ft 2

2. The top view of a two-tiered wedding cake is shown. Each tier is a regular hexagon. What percent of the bottom tier is covered by the top tier? Round to the nearest percent.

bottom tier

top tier

5 in.8 in.

39%

4. A yield sign is a regular triangle and is available in two sizes: 30 inches or 36 inches. Find how much more metal is needed to make a 36 inch sign than a 30 inch sign. Answer to the nearest percent.

YIELD

30 in. or36 in.

44%

Choose the best answer.

5. A regular hexagon has an apothem of 4.6 centimeters. Which is the best estimate for the area of the hexagon?

A 36.7 cm2

B 63.5 cm2

C 73.3 cm2

D 146.6 cm2

7. A cyclist travels 50 feet after 7.34 rotations of her bicycle wheels. What is the approximate diameter of the wheels?

A 13 in. C 26 in.

B 24 in. D 28 in.

6. An amusement park ride is made up of a large circular frame that holds 50 riders. The circumference of the frame is about 138 feet. What is the diameter of the ride to the nearest foot?

F 22 ft H 69 ft

G 44 ft J 138 ft

8. A regular pentagon has side length 16 inches. What is the area of the pentagon to the nearest square inch?

F 440 in2 H 544 in2

G 369 in2 J 881 in2

9-2


Name Date Class

LESSON

9-2Reading StrategiesVocabulary Development

The diagram below describes the parts of a regular polygon and how to use those parts to find the area of the polygon.

Answer the following.

1. What is true about the sides of a regular polygon?

They have the same length.

2. How many central angles are in a regular octagon? 8

3. What is the perimeter of a regular nonagon with side length 8 feet? P � 72 ft

Use the formula for the area of a regular polygon to find the area of each figure. Round to the nearest tenth if necessary.

4.

��

��

5. ��

��

��

A � 498.8 cm2 A � 196 in2

The center of a regular polygon is equidistant from the vertices.

The apothem is the distance from the center to the side.

A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. The sum of the measures of all the central angles equals 360°.

The area of a regular polygon with apothem a and perimeter P is:

A � 1__2

aP

Example:

Each side of a regular hexagon is 10 m. The apothem is 5�

� 3 m. Find the area.

• First find the perimeter.

P � 6(10) � 60 m

• Then substitute into the formula.

A � 1__2

(5 ��

3)(60) � 259.8 m2

A regular polygon has all sides of equal length and all interior angles of equal measure.

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