8.7 Circumference and Area of Circles

Page 1 of 8

8.7

Goal

Find the circumference and

area of circles.

Key Words

Circumference and Area

of Circles

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.

? circle

radius

? center

The distance across the circle, through the

center, is the diameter . The diameter d is

twice the radius r. So, d 5 2r.

? radius

? diameter

? circumference

diameter

The circumference of a circle is the distance

around the circle.

? central angle

? sector

center

circumference

For any circle, the ratio of the circumference to its diameter is denoted

by the Greek letter ¦Ð, or pi. The number ¦Ð is 3.14159 . . . , which is an

irrational number. This means that ¦Ð neither terminates nor repeats.

So, an approximation of 3.14 is used for ¦Ð.

CIRCUMFERENCE OF A CIRCLE

Words

Circumference 5 ¦Ð(diameter)

5 2¦Ð(radius)

Symbols

r

C 5 ¦Ðd or C 5 2¦Ðr

EXAMPLE

1

Find the Circumference of a Circle

Find the circumference of the circle.

4 in.

Solution

Student Help

C 5 2¦Ðr

STUDY TIP

When simplifying an

expression involving ¦Ð,

substitute 3.14 for ¦Ð.

You can also use the ¦Ð

key on your calculator,

as in Examples 2 and 3.

452

Chapter 8

5 2¦Ð(4)

Substitute 4 for r.

5 8¦Ð

Simplify.

¡Ö 8(3.14)

Use 3.14 as an approximation for ¦Ð.

5 25.12

Multiply.

ANSWER

Polygons and Area

Formula for the circumference

? The circumference is about 25 inches.

Page 2 of 8

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

16 in.

9 ft

AREA OF A CIRCLE

Words

Area 5 ¦Ð(radius)2

Symbols

A 5 ¦Ðr

EXAMPLE

2

r

2

Find the Area of a Circle

Find the area of the circle.

7 cm

Student Help

KEYSTROKE HELP

If your calculator has

¦Ð written above a key,

use the following

keystrokes to simplify

49¦Ð:

49

Solution

A 5 ¦Ðr 2

5 ¦Ð(7)

Formula for the area of a circle

2

Substitute 7 for r.

5 49¦Ð

Simplify.

¡Ö 153.94

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.

r

A 5 380 ft2

Solution

A 5 ¦Ðr 2

380 5 ¦Ðr

120.96 ¡Ö r

11 ¡Ö r

ANSWER

2

2

Formula for the area of a circle

Substitute 380 for A.

Divide each side by ¦Ð. Use a calculator.

Take the positive square root.

? The radius is about 11 feet.

8.7

Circumference and Area of Circles

453

Page 3 of 8

Find the Area of a Circle

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

number.

4.

5.

6.

8 in.

12 ft

3 cm

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.

Central Angles An angle whose vertex is the center

of a circle is a central angle of the circle.

sector

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.

central

angle

Measure of central angle

Area of sector

}}} 5 }}}

Area of entire circle

Measure of entire circle

EXAMPLE

4

Find the Area of a Sector

Find the area of the blue sector.

9m

1208

Solution

1 First find the area of the circle.

¡ñ

A 5 ¦Ðr 2 5 ¦Ð(9)2 ¡Ö 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.

¡ñ

Area of sector

Measure of central angle

}}} 5 }}}

Area of entire circle

Measure of entire circle

Visualize It!

A circle contains two

straight angles. So,

there are 1808 1 1808,

or 3608, in a circle.

1808

x

1208

}} 5 }}

254

3608

3608

or

Substitute.

360x 5 254 p 120

Cross product property

360x 5 30,480

Simplify.

3 6 0x

30,480

} } 5 }}

360

360

Divide each side by 360.

1808

x ¡Ö 84.67

ANSWER

454

Chapter 8

Polygons and Area

Simplify.

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

Page 4 of 8

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 5 22 m2

x

1808

}} 5 }}

?

?

8. A 5 28 ft2

x

?

}} 5 }}

?

3608

1808

1708

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

whole number.

9.

10.

11.

5 cm

908

2 ft

608

6 in.

1358

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

3.

?

14 in.

4.

11 cm

?

5.

3.5 m

?

6.

?

1 ft

r

d

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

the missing number.

¡ö

7. C 5 2¦Ð ?

¡ö

8. A 5 ? (3)2

¡ö

9. A 5 ¦Ð( ? )2

8

3

8.7

14

Circumference and Area of Circles

455

Page 5 of 8

Practice and Applications

Extra Practice

Finding Circumference Find the circumference of the circle. Round

your answer to the nearest whole number.

See p. 690.

10.

11.

12.

7 cm

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.

16.

4 ft

18.

17.

6 in.

1 cm

19.

20.

19 yd

16 ft

9m

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 5 ¦Ðr 2

¡Ö (3.14)(12)2

12

5 452.16

Homework Help

Example 1:

Example 2:

Example 3:

Example 4:

456

Exs. 10¨C14

Exs. 15¨C20

Exs. 24¨C29

Exs. 31¨C35

Chapter 8

23.

You be the Judge

One of your

classmates states that if the radius of a

circle is doubled, then its area doubles.

Do you agree or disagree? Explain your

reasoning using the circles at the right.

Polygons and Area

5

10

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