LESSON

Date

Class

Reteach

Developing Formulas for Circles and Regular Polygons

Circumference and Area of Circles

9-2

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

Find the circumference of circle S in which A Step 1 Use the given area to solve for r. A 81 cm 2

2 81 cm .

r2 r r2 r

2

Area of a circle Substitute 81 for A. Divide both sides by . Take the square root of both sides. cm

81 cm2 9 cm Step 2

Use the value of r to find the circumference. C C 2 r 2 (9 cm) 18 cm Circumference of a circle 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

5m

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.

18 cm

5. the circumference of circle X in 2 which A 49 in

6. the radius of circle Y in which C

C

Copyright © by Holt, Rinehart and Winston. All rights reserved.

14 in.

14

r

9 cm

Holt Geometry

Name

LESSON

Date

Class

Reteach

Developing Formulas for Circles and Regular Polygons continued Area of Regular Polygons The center is equidistant from the vertices.

9-2

The area of a regular polygon with apothem a and perimeter P 1 is A __aP. 2

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

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 360° angle measure of a regular n-gon is ____. n A central angle has its vertex at the center. This central angle measure is 360 ____ 60 . 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° tan 30° leg opposite 30° angle ____________________ leg adjacent to 30° angle 5 cm _____ Write a tangent ratio. Substitute the known values. Solve for a.

Step 3

a 5 cm a ______ tan 30° Use the formula to find the area. 1 A __ aP 2 A A 5 1 __ ______ 60 2 tan 30 259.8 cm 2

a

5 ______ , P tan 30°

6

10 or 60 cm

Simplify.

Find the area of each regular polygon. Round to the nearest tenth. 7. 12 cm

8.

4 in.

A

695.3 cm2 31.2 m2

15

A

58.1 in2 377.0 ft2

Holt Geometry

9. a regular hexagon with an apothem of 3 m

10. a regular decagon with a perimeter of 70 ft

A

A

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Name

LESSON

Date

Class

Name

LESSON

Date

Class

Developing Formulas for Circles and Regular Polygons In Exercises 1–3, fill in the blanks to complete each formula. 1 __ aP 1. The area of a regular polygon with apothem a and perimeter P is A � 2 �d 2. A circle with diameter d has circumference C � . �r 2 3. A circle with radius r has area A � . Use the area and circumference formulas for circles to find each measurement. Give your answers in terms of �. 4. 5 ft

9-2

Practice A

9-2

Practice B

Developing Formulas for Circles and Regular Polygons

2.

25 m

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

.

1.

�

� 4� in.

the area of �V

the area of �H

5.

�

20 in. �

A � 625� m2

3.

�

(� � � ) yd

A � 4a 2� in2

4.

�

1200 mi

the area of �A

the area of �Q

A � 25� ft2

6.

� 18 cm

A � 100� in2

7.

13 mi

the circumference of �M

the circumference of �R

C � (2x � 2y)� yd

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

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

2

�

C � 1200� mi r � � cm

d � (2x � 2) km

the circumference of �W

the circumference of �N

C � 18� cm

8. the radius of �I in which A � 144� meters

2

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

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

C � 26� mi 12 m 2 km 256� yd2

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