r

C (base)12

C 12

(height)

r

MAIN IDEAFind the areas of circles.

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11-4

• Fold a paper plate in half four times to

divide it into 16 equal-sized sections.

• Label the radius r as shown. Let C

represent the circumference of the circle.

• Cut out each section; reassemble to

form a parallelogram-shaped figure.

1. What is the measurement

of the base and the height? 1 _

2 C ; r

2. Substitute these values into

the formula for the area of a

parallelogram. A = 1 _

2 C (r )

3. Replace C with the expression for the circumference of a circle, 2π r.

Simplify the equation and describe what it represents.

In the Mini Lab, the formula for the area of a parallelogram was used to

develop a formula for the area of a circle.

Key ConceptArea of a CircleWords The area A of a circle equals the

product of π and the square of its radius r.

Model r

Symbols A = π r 2

Find the Area of a Circle

1 Find the area of the circle.

A = π r 2 Area of a circle

A = π · 2 2 Replace r with 2.

[π] 2 12.56637061

2 in.

The area of the circle is approximately 12.6 square inches.

a. Find the area of a circle with a radius of 3.2 centimeters. Round to

the nearest tenth. 32.2 cm 2

3. A = 1 _

2 (2πr ) (r ); A = πr 2 ; the area of a circle3. A =

1 _

2 (2πr ) (r ); A = πr 2 ; the area of a circle

Area of Circles

Lesson 11-4 Area of Circles 589

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Identifying What is Given Before finding area, be sure to read the question carefully and identify if the radius or diameter is given.

2 COINS Find the area of the face of the Wisconsin quarter shown.

The diameter of the quarter is 24 millimeters, 24 mm

so the radius is 1 _ 2 (24) or 12 millimeters.

A = π r 2 Area of a circle

A = π · 12 2 Replace r with 12.

A ≈ 452.4 Use a calculator.

The area is approximately 452.4 square

millimeters.

b. POOLS The bottom of a circular swimming pool with diameter

30 feet is painted blue. How many square feet are blue? 706.5 ft 2

A sector of a circle is a region of a circle bounded by two radii.

3 Ellis draws a circle with a diameter of

16 in.

180°

16 inches, and shades one region of the

circle. Find the approximate area of

the sector.

A 100 in 2 C 402 in 2

B 201 in 2 D 804 in 2

Read the ItemThe diameter of the circle is 16 inches. Since there are 360° in

a circle, the sector is 180° _

360° or

1 _

2 the area of the entire circle.

Solve the ItemA = π r 2 Area of a circle

A = π · 8 2 Replace r with 16 ÷ 2 or 8.

A ≈ 200 Multiply. Use 3.14 for π.

The area of the sector is approximately 1 _ 2 (200) or 100 square inches.

The answer is A.

c. Ray drew one circle with a radius of 7 centimeters and another

circle with a radius of 10 centimeters. Find the approximate

difference between the areas of the circles. H

F 28 cm 2 G 40 cm 2 H 160 cm 2 J 254 cm 2

Radii Radii The plural form of radius is radii.

590 Chapter 11 Measurement: Two- and Three-Dimensional Figures

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Examples 1, 2(pp. 589–590)

Find the area of each circle. Round to the nearest tenth.

1.5 cm

2.

9 in.

3. diameter = 16 m

4. diameter = 13 ft

Example 3(p. 590)

5. MULTIPLE CHOICE Kenneth draws the circle

14 ydshown at the right. He shades one region

of the circle. What is the approximate area

of the sector? B

A 88 yd 2 C 310 yd 2

B 154 yd 2 D 615 yd 2

Find the area of each circle. Round to the nearest tenth.

6.

8 cm

7.3 in.

8.

11 ft

9.17 cm

10.2.4 m

11.

3.2 mm

12. diameter = 8.4 m 13. diameter = 12.6 cm 14. radius = 4 1 _ 2 in.

15. radius = 3 3 _ 4 ft 16. diameter = 9 1 _

4 mi 17. diameter = 20 3 _

4 yd

18. PATCHES Find the area of the Girl Scout patch

shown if the diameter is 1.25 inches. Round

to the nearest tenth. 1. 2 in 2

19. TOOLS A sprinkler that sprays water in a

circular area can be adjusted to spray up to

30 feet. To the nearest tenth, what is the

maximum area of lawn that can be watered

by the sprinkler? 2,827.4 ft 2

ESTIMATION Estimate to find the approximate area of each circle.

20. 8 cm

21. 5.9 ft

22. 13.8 in.

78.5 cm 2 78.5 cm 2 254.5 in 2 254.5 in 2 201.1 m 2 201.1 m 2

132.7 ft 2 132.7 ft 2

Exercise Levels

A: 6–19

B: 20–29

C: 30–35

Exercise Levels

A: 6–19

B: 20–29

C: 30–35

For Exercises

6–7, 10–11, 14–15, 19

8–9, 12–13, 16–1836–38

See Examples

1

2

3

HOMEWORK HELPHELP

For Exercises

6–7, 10–11, 14–15, 19

8–9, 12–13, 16–1836–38

See Examples

1

2

3

HOMEWORK HELPHELP 201.1 cm 2 201.1 cm 2 28.3 in 2 28.3 in 2 95.0 ft 2 95.0 ft 2

227.0 cm 2 227.0 cm 2 18.1 m 2 18.1 m 2 32.2 mm 2 32.2 mm 2

55.4 m 2 55.4 m 2 124.7 cm 2 124.7 cm 2 63.6 in 2 63.6 in 2

44.2 ft 2 44.2 ft 2 67.2 mi 2 67.2 mi 2 338.2 yd 2 338.2 yd 2

Sample

answer:

3 × 4 2 =

48 cm 2

Sample

answer:

3 × 4 2 =

48 cm 2

Sample

answer:

3 × 6 2 =

108 ft 2

Sample

answer:

3 × 6 2 =

108 ft 2

Sample

answer:

3 × 7 2 =

147 in 2

Sample

answer:

3 × 7 2 =

147 in 2

★ indicates multi-step problem

Answers were computed using the π key on a calculator.

Lesson 11-4 Area of Circles 591

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12 ft4 ft

See pages 698, 714.

EXTRA PRACTICEPRACTICE

For Exercises 23–26, use a compass to draw the circle

shown on centimeter grid paper.

23. Count the number of squares that lie completely

within the circle. Then count the number of squares

that lie completely within or contain the circle.

24. Estimate the area of the circle by finding the mean

of the two values you found in Exercise 23.

25. Find the area of the circle by using the area formula.

26. How do the areas you found in Exercises 24 and 25 compare to one another?

27. A semicircle is half a circle. Find the area of the

8.6 m

semicircle to the nearest tenth.

28. Which has a greater area, a triangle with a base

of 100 feet and a height of 100 feet or a circle with

diameter of 100 feet? Justify your selection.

29. RADIO SIGNALS A radio station sends a signal in a circular area with an 80-

mile radius. Find the approximate area in square kilometers that receives

the signal. (Hint: 1 square mile ≈ 2.6 square kilometers)

H.O.T. Problems 30. REASONING If the length of the radius of a

circle is tripled, does the area also triple?

Explain your reasoning.

CHALLENGE Find the area of the shaded region in each figure. Round

to the nearest tenth.

31.

12 m

8 m

32.

5.25 in.

33.

12.5 cm

3.5 cm

34. FIND THE ERROR Dasan and Carmen are finding the area of a circle

that has a diameter of 16 centimeters. Who is correct? Explain.

35. MATHWRITING IN Write and solve a real-world problem in which you

would solve the problem by finding the area of a circle.

Dasan Carmen

A = π (16) 2 ≈ 804 cm 2 A = π (8) 2

≈ 201 cm 2

592 Chapter 11 Measurement: Two- and Three-Dimensional Figures

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36. The radius of the half dollar in

centimeters is given below. Find

the approximate area of the shaded

sector. A

1.95 cm

A 6 cm 2 C 14 cm 2

B 12 cm 2 D 28 cm 2

37. Which equation could be used to find

the area in square inches of a circle

with a radius of 12 inches? J

F A = 6 × π H A = 12 × π

G A = π × 6 2 J A = π × 12 2

38. Which two figures have the same area

shaded? A

8 m

12 m

Figure I

7.5 m

12 m

Figure II

180°

12 m

Figure III Figure IV

6 m

10 m

12 m

A Figure I and Figure IV

B Figure I and Figure II

C Figure II and Figure IV

D Figure II and Figure III

39. MEASUREMENT What is the circumference of a circle that has a radius of

8 yards? Use 3.14 for π and round to the nearest tenth if necessary.

(Lesson 11-3) 50.2 yd

40. MEASUREMENT Find the area of a triangle with a base of 21 meters and

a height of 27 meters. (Lesson 11-2) 283.5 m 2

Find the area of each parallelogram. Round to the nearest tenth if necessary.

(Lesson 11-1)

41.

12 in.

10 in.

42.

7.9 cm

5 cm

43.

11.5 m

8.7 m

PREREQUISITE SKILL Simplify each expression. (Lessons 1-2, 1-3, and 1-4)

44. 8. 5 2 72.25 45. 3.14 · 6 2 113.04 46. 1 _ 2 · 5.4 2 + 11 25.58 47. 1 _

2 · 7 2 + (9) (14) 150.5

120 in 2 120 in 2

39.5 cm 2 39.5 cm 2 100.1 m 2 100.1 m 2

Lesson 11-4 Area of Circles 593

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