11.4 Circumference and Arc Length
2/17/2011

Objectives


Find 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 length

The 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
Circle

The circumference C of a circle is C =  d or
C = 2  r, where d is the diameter of the circle and r is the radius of the circle.
diameter d

Ex. 1: Using circumference

Find
(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 box

Find 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 Length

In 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 °.

Arc length of AB = m

AB
360 °
P
• 2  r
A
B

More . . .

The length of a semicircle is half the circumference, and the length of a 90 ° arc is one quarter of the circumference. d
½ • 2  r r
¼ • 2  r

Ex. 2: Finding Arc Lengths

Find the length of each arc.
a.
5 cm
50 °
A
B c.
E
7 cm
100 °
F

Ex. 2: Finding Arc Lengths

Find the length of each arc.
b.
C 7 cm
50 °
D


Ex. 2: Finding Arc Lengths
Find the length of each arc.
E c.
7 cm
100 °
F


Ex. 2: Finding Arc Lengths
Find the length of each arc.
a.
5 cm
50 °
A
B

# of °
360 °

50 °
360 °
• 2  r
• 2 
(5)

4.36 centimeters


Ex. 2: Finding Arc Lengths
Find the length of each arc.
b.
C 7 cm
50 °
D

# of °
360 °

50 °
360 °
• 2  r
• 2 
(7)

6.11 centimeters


Ex. 2: Finding Arc Lengths
Find the length of each arc.
c.
E
7 cm
100 °
F

# of °
360 °

100 °
360 °
• 2  r
• 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 Revolutions


The 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)