11.4 Circumference and Arc Length

2/17/2011

ObjectivesFind the circumference of a circle

and the length of a circular arc.Use circumference and arc length

to solve real-life problems.

Finding circumference and arc lengthThe circumference of a circle is

the distance around the circle.

For all circles, the ratio of the circumference to the diameter is the same. This ratio is known as or pi.

Theorem 11.6: Circumference of a CircleThe

circumference C of a circle is C = d or C = 2r, where d is the diameter of the circle and r is the radius of the circle.

diameter d

Ex. 1: Using circumferenceFind

(Round decimal answers to two decimal places)

◦(a) the circumference of a circle with radius 6 centimeters and

◦(b) the radius of a circle with circumference 31 meters.

What’s the difference??Find the exact radius of a circle

with circumference 54 feet.

Find the radius of a circle with circumference 54 feet.

Extra Examples Write below previous boxFind the exact circumference of a

circle with diameter of 15.

Find the exact radius of a circle with circumference of 25.

And . . . An arc length is a portion of the circumference of a circle.

You can use the measure of an arc (in degrees) to find its length (in linear units).

Finding the measure of an Arc LengthIn a circle, the

ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360°.

P

B

A

Arc length of =360°

• 2r AB

mAB

More . . .

The length of a semicircle is half the circumference, and the length of a 90° arc is one quarter of the circumference.

½ • 2r

r¼ • 2r

d

Ex. 2: Finding Arc Lengths

Find the length of each arc.

5 cm

B

A

50°

a. 7 cm

F

E

100°c.

7 cm

D

C

b.

50°

Ex. 2: Finding Arc LengthsFind the length of each arc.

7 cm

F

E

100°

c.

Ex. 2: Finding Arc LengthsFind the length of each arc.

Ex. 2: Finding Arc Lengths

AB

Find the length of each arc.

5 cm

B

A

50°

a.

a. Arc length of = AB 50°

360°• 2(5)

a. Arc length of = # of °

360°• 2r

4.36 centimeters

Ex. 2: Finding Arc LengthsFind the length of each arc.

7 cm

D

C

50°

b. b. Arc length of = CD # of °

360°• 2r

b. Arc length of = CD 50°

360°• 2(7)

6.11 centimeters

Ex. 2: Finding Arc Lengths

EF

Find the length of each arc.

7 cm

F

E

100°

c. c. Arc length of = # of °

360°• 2r

c. Arc length of = EF 100°

360°• 2(7)

12.22 centimeters

In parts (a) and (b) in Example 2, note that the arcs have the same measure but different lengths because the circumferences of the circles are not equal.

Ex. 3: Tire RevolutionsThe dimensions of a

car tire are shown.To the nearest foot,

how far does the tire travel when it makes 8 revolutions?

P. 214 (1-10)

Assignment