can also be measured in length, as a portion of the circumference.
Arc Length:
The length of an arc or a portion of a circle’s circumference.
The arc length is directly related to the degree arc measure.
Example 5:
Find the length of
c
PQ
. Leave your answer in terms of
π
.
Solution:
In the picture, the central angle that corresponds with
c
PQ
is
60
◦
. This means that
m
c
PQ
= 60
◦
.
Think of the arc length as a portion of the circumference. There are
360
◦
in a circle, so
60
◦
would be
1
6
of
that
(
60
◦
360
◦
=
1
6
)
. Therefore, the length of
c
PQ
is
1
6
of the circumference.
length of
c
PQ
=
1
6
·
2
π
(9) = 3
π
Arc Length Formula:
The length of
c
AB
=
m
c
AB
360
◦
·
π
d
or
m
c
AB
360
◦
·
2
π
r
.
Another way to write this could be
x
◦
360
◦
·
2
π
r
, where
x
is the central angle.
Example 6:
The arc length of
c
AB
= 6
π
and is
1
4
the circumference. Find the radius of the circle.
Solution:
If
6
π
is
1
4
the circumference, then the total circumference is
4(6
π
) = 24
π
. To find the radius,
plug this into the circumference formula and solve for
r
.
24
π
= 2
π
r
12 =
r
Example 7:
Find the measure of the central angle or
c
PQ
.
Solution:
Let’s plug in what we know to the Arc Length Formula.
318

15
π
=
m
c
PQ
360
◦
·
2
π
(18)
15 =
m
c
PQ
10
◦
150
◦
=
m
c
PQ
Example 8:
The tires on a compact car are 18 inches in diameter. How far does the car travel after the
tires turn once? How far does the car travel after 2500 rotations of the tires?
Solution:
One turn of the tire is the circumference. This would be
C
= 18
π
≈
56
.
55
in
. 2500 rotations
would be
2500
·
56
.
55
in
= 141371
.
67
in
, 11781 ft, or 2.23 miles.
Know What? Revisited
The entire length of the crust, or the circumference of the pizza is
14
π
≈
44
in
.
In
1
8
of the pizza, one piece would have
44
8
≈
5
.
5
inches of crust.
Review Questions
• Questions 1-10 are similar to Examples 1 and 2.
• Questions 11-14 are similar to Examples 3 and 4.
• Questions 15-20 are similar to Example 5.
• Questions 21-23 are similar to Example 6.
• Questions 24-26 are similar to Example 7.
• Questions 27-30 are similar to Example 8.
Fill in the following table. Leave all answers in terms of
π
.
Table 6.1:
diameter
radius
circumference
1.
15
2.
4
3.
6
4.
84
π
5.
9
6.
25
π
7.
2
π
8.
36
9. Find the radius of circle with circumference 88 in.
319

10. Find the circumference of a circle with
d
=
20
π
cm
.
Square
PQS R
is inscribed in
⊙
T
.
RS
= 8
√
2
.
11. Find the length of the diameter of
⊙
T
.
12. How does the diameter relate to
PQS R
?
13. Find the perimeter of
PQS R
.
14. Find the circumference of
⊙
T
.
Find the arc length of
c
PQ
in
⊙
A
. Leave your answers in terms of
π
.
15.
16.
17.
18.
320

19.
20.
Find
PA
(the radius) in
⊙
A
. Leave your answer in terms of
π
.
21.
22.
23.
Find the central angle or
m
c
PQ
in
⊙
A
. Round any decimal answers to the nearest tenth.
24.
321

25.
26.
For questions 27-30, a truck has tires with a 26 in diameter.
27. How far does the truck travel every time a tire turns exactly once? What is this the same as?