8.7 Circumference and Area of Circles

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8.7 Circumference and Area

of Circles

Goal

Find the circumference and area of circles.

Key Words

? circle ? center ? radius ? diameter ? circumference ? central angle ? sector

A circle is the set of all points in a plane that are the same distance from a given point, called the center of the circle. A circle with center P is called "circle P," or P.

The distance from the center to a point on the circle is the radius . The plural of radius is radii.

The distance across the circle, through the center, is the diameter . The diameter d is twice the radius r. So, d 2r.

radius diameter

center

The circumference of a circle is the distance around the circle.

circumference

For any circle, the ratio of the circumference to its diameter is denoted by the Greek letter , or pi. The number is 3.14159 . . . , which is an irrational number. This means that neither terminates nor repeats. So, an approximation of 3.14 is used for .

CIRCUMFERENCE OF A CIRCLE

Words Circumference (diameter)

2(radius)

r

Symbols C d or C 2r

EXAMPLE 1 Find the Circumference of a Circle

Find the circumference of the circle. 4 in.

Student Help

STUDY TIP When simplifying an expression involving , substitute 3.14 for . You can also use the key on your calculator, as in Examples 2 and 3.

Solution C 2r 2(4) 8 8(3.14) 25.12

Formula for the circumference Substitute 4 for r. Simplify.

Use 3.14 as an approximation for .

Multiply.

ANSWER The circumference is about 25 inches.

452 Chapter 8 Polygons and Area

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Find the Circumference of a Circle

Find the circumference of the circle. Round your answer to the nearest whole number.

1.

2.

3.

6 cm 9 ft

16 in.

AREA OF A CIRCLE

Words Area (radius)2

r

Symbols A r2

Student Help

KEYSTROKE HELP If your calculator has written above a key, use the following keystrokes to simplify 49:

49

EXAMPLE 2 Find the Area of a Circle

Find the area of the circle. 7 cm

Solution A r 2 (7)2 49 153.94

Formula for the area of a circle Substitute 7 for r. Simplify. Use a calculator.

ANSWER The area is about 154 square centimeters.

EXAMPLE 3 Use the Area of a Circle

Find the radius of a circle with an area of 380 square feet.

Solution

r A 380 ft2

A r 2 Formula for the area of a circle

380 r 2 Substitute 380 for A.

120.96 r 2

Divide each side by . Use a calculator.

11 r

Take the positive square root.

ANSWER The radius is about 11 feet.

8.7 Circumference and Area of Circles 453

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Find the Area of a Circle

Find the area of the circle. Round your answer to the nearest whole number.

4. 8 in.

5. 3 cm

6. 12 ft

Student Help

VOCABULARY TIP The term radius is also used to name a segment that connects the center of a circle to a point on the circle. Two such radii are used to determine a sector of a circle.

Visualize It!

A circle contains two straight angles. So, there are 180 180, or 360, in a circle.

180

360

or

180

Central Angles An angle whose vertex is the center of a circle is a central angle of the circle.

A region of a circle determined by two radii and a part of the circle is called a sector of the circle.

Because a sector is a portion of a circle, the following proportion can be used to find the area of a sector.

Area of sector

Measure of central angle

Area of entire circle Measur e of entir e circle

sector

central angle

EXAMPLE 4 Find the Area of a Sector

Find the area of the blue sector.

9 m 120

Solution

1 First find the area of the circle.

A r 2 (9)2 254.47

The area of the circle is about 254 square meters.

2 Then find the area of the sector. Let x equal the area of the sector.

Area of sector

Measure of central angle

Area of entire circle Measur e of entir e circle

25x4 132600

Substitute.

360x 254 p 120 Cross product property

360x 30,480

Simplify.

3366 00x 303,64 080

Divide each side by 360.

x 84.67

Simplify.

ANSWER The area of the sector is about 85 square meters.

454 Chapter 8 Polygons and Area

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Find the Area of a Sector

In Exercises 7 and 8, A represents the area of the entire circle and x represents the area of the blue sector. Complete the proportion used to find x. Do not solve the proportion.

7. A 22 m2

8. A 28 ft2

x? 18?0

180

x? 36?0

170

Find the area of the blue sector. Round your answer to the nearest whole number.

9.

2 ft 60

10. 5 cm

90

11.

6 in. 135

8.7 Exercises

Guided Practice

Vocabulary Check

1. Sketch a circle. Sketch and label a radius and a diameter of the circle.

2. Describe how to find the circumference of a circle given its radius.

Skill Check Copy and complete the table below.

Radius, r Diameter, d

r

3.

?

14 in.

4. 11 cm

?

d

5.

3.5 m

?

6.

?

1 ft

Write an equation for the area A or the circumference C by filling in

the missing number.

7. C 2 ?

8. A ? (3)2

9. A ( ? )2

8

3

14

8.7 Circumference and Area of Circles 455

Page 5 of 8

Practice and Applications

Extra Practice

See p. 690.

Finding Circumference Find the circumference of the circle. Round your answer to the nearest whole number.

10. 7 cm

11.

12.

10 m

2 in.

13. A circle with a radius of 13 yards

14. A circle with a diameter of 15 meters

Finding Area Find the area of the circle. Round your answer to the nearest whole number.

15. 4 ft

16. 1 cm

17. 6 in.

18. 9 m

19. 16 ft

20. 19 yd

21. Cranberries To harvest cranberries, the field is flooded so that the berries float. What area of cranberries can be gathered into a circular region with a radius of 5.5 meters?

22. Error Analysis A student was asked to find the area of the circle below. Describe any errors.

A r2 (3.14)(12)2 12 452.16

Homework Help

Example 1: Exs. 10?14

23. You be the Judge One of your

classmates states that if the radius of a

5

Example 2: Exs. 15?20 Example 3: Exs. 24?29

circle is doubled, then its area doubles.

10

Example 4: Exs. 31?35

Do you agree or disagree? Explain your

reasoning using the circles at the right.

456 Chapter 8 Polygons and Area

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