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Chapter 9

Geometry

CHAPTER

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9Geometry

9.1 Perimeter

9.2 Area

9.3 Circles

9.4 Volume

9.5 Angles and Triangles

9.6 Square Roots and the Pythagorean Theorem

OBJECTIVES

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9.3 Circles

a Find the length of a radius of a circle given the length of a diameter, and find the length of a diameter given the length of a radius.

b Find the circumference of a circle given the length of a diameter or a radius.

c Find the area of a circle given the length of a diameter or a radius.

d Solve applied problems involving circles.

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Segment is a diameter. A diameter is a segment that passes through the center of the circle and has endpoints on the circle.

Segment is called a radius. A radius is a segment with one endpoint on the center and the other endpoint on the circle.

OB

AC

r

B

A

C

Od

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Diameter and RadiusSuppose that d is the diameter of a circle and r is the radius. Then

d = 2 r and .2

dr

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Example

Find the length of a radius of this circle.

Solution

2

dr

2

16 m

16 m

8 mThe radius is 8 m.

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Example

Find the length of a diameter of this circle.

Solution

2d r

f21

t2

½ ft

1 ftThe diameter is 1 ft.

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The perimeter of a circle is called its circumference.

Circumference and DiameterThe circumference C of a circle of diameter d is given by

The number is about 3.14, or about

C d

22

.7

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Example

Find the circumference of this circle. Use 3.14 for .

Solution

8 m

C d

3.14 8 cm

25.12 cm

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Circumference and RadiusThe circumference C of a circle of radius r is given by

2 .C r

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Example

Find the circumference of this circle. Use 22/7 for .

Solution140 in.2C r

140 in22

2 .7

140i2 2 .n2

7

44 20 in. 880 in.

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Example

Find the perimeter of this figure. Use 3.14 for .

Solution

We let P = the perimeter.

Since there is 1/2 a circle replacing the fourth side of a square, we add half the circumference to the lengths of the three line segments.

5.2 km

8.2 km

5.6 km

8.2 5.6 8.21

22

5.2P 22 3.14 5.2

22 16.328 38.328 km

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Area of a CircleThe area of a circle with radius of length r is given by

2 or .A r r A r r

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Example

Find the area of this circle. Use 3.14 for . Round to the nearest hundredth.

Solution 3.6 m

A r r

3.6 m 3.6 m

23.14 12.96 m

240.6944 m 240.69 m

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Example

A local pizza parlor is ordering new square serving plates. If they order a 16 in. plate how much area will show when a 14 in. diameter pizza is placed on the pan?

Familiarize. Look at the drawing, notice

that the corners of the box will be visible when the pizza is in place. We let A = the area of the box visible.

14 in.

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continued

Translate.

Area of box minus Area of pizza is Visible Area

s s – r r = A

Solve. The radius is ½ the diameter, or 7 in.

s s – r r = A

16 in. 16 in. – 3.14 7 in. 7 in. A

256 in2 – 153.86 in2 A

102.14 in2 A

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continued

Check. We can check by repeating our calculations. Note also that the area of the plate does not exceed the area of the pizza.

State.

When the pizza is in place, about 102.14 in2 of plate will be visible.