16.2 Arc Length and Radian Measure

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Name
Class
Date
16.2 Arc Length and Radian Measure
Essential Question: How do you find the length of an arc?
Explore 1
Deriving the Formula for Arc Length
Resource
Locker
An arc is an unbroken part of a circle consisting of two points called the endpoints and all
the points on the circle between them. Arc length is understood to be the distance along a
circular arc measured in linear units (such as feet or centimeters). You can use proportional
reasoning to find arc length.
⁀. Express your answer in terms of π and rounded to the nearest tenth.
Find the arc length of AB
60°
B
A
9 cm
A
First find the circumference of the circle.
C = 2πr
Substitute the radius, 9, for r.
C = 18π
B
60 or _
1 of the circumference.
The entire circle has 360°. Therefore, the arc’s length is _
360
6
Fill in the blanks to find the arc length.
© Houghton Mifflin Harcourt Publishing Company
1 ⋅ 18π
⁀= _
Arc length of AB
6
3π
=
≈
1 of the circumference.
Arc length is _
6
Multiply.
9.4
⁀ is
So, the arc length of AB
Use a calculator to evaluate. Then round.
3π cm
or
9.4 cm
.
Reflect
1.
How could you use the reasoning process you used above to find the length of an arc
of the circle that measures m°?
m
m
m
of the circumference, or ___
⋅ 18π. This can be written as __
π.
The arc length is ___
360
360
20
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Explain 1
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Applying the Formula for Arc Length
You were able to find an arc length using concepts of circumference. Using the same
reasoning results in the formula for finding arc length.
Arc Length
m ⋅ 2πr.
The arc length, s, of an arc with measure m° and radius r is given by the formula s = _
360
A
m
r
B
Example 1 Find the arc length.
 On a clock face, the minute hand of a clock is 10 inches long. To the nearest
tenth of an inch, how far does the tip of the minute hand travel as the time
progresses from 12:00 to 12:15?
The minute hand moves 15 minutes.
15 minutes = _
1 ⋅ 360° = 90°.
1 so the central angle formed is_
_
4
60 minutes 4
m ⋅ 2πr
Use the formula for arc length.
s=_
360
90 ⋅ 2π(10)
Substitute 10 for r and 90 for m.
=_
360
Simplify.
= 5π
Simplify.
≈ 15.7 in.
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 The minute hand of a clock is 6 inches long. To the nearest tenth of an inch, how far
does the tip of the minute hand travel as the time progresses from 12:00 to 12:30?
The minute hand moves 30 minutes.
1
30 minutes = __
1
_
_
2 , so the central angle formed is 2 ⋅ 360° = 180° .
60 minutes
m ⋅ 2πr
Use the formula for arc length.
s=_
360
Substitute 3 for r and 35 for m.
180
3
_
⋅ 2π
=
360
Simplify.
= 3π
Simplify.
≈ 9.4 in.
( )
The length of the arc is
9.4 inches.
Reflect
2.
Discussion Why does the formula represent the length of an arc of a circle?
Suppose a circle has a radius r. Find the length of an arc of the circle that measures m°.
m
of the circumference of the circle. The circumference is
Since a circle is 360°, each arc is ___
360
m
2πr, so the arc length is ___
⋅ 2πr.
360
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Your Turn
3.
The minute hand of a clock is 8 inches long. To the nearest tenth of an inch, how far
does the tip of the minute hand travel as the time progresses from 12:00 to 12:45?
3
3
45 minutes _
__
= ,so _ ⋅ 360° = 270°
4
4
60 minutes
m
_
⋅ 2πr
s=
360
270
= _ ⋅ 2π(8) ≈ 37.7 in.
360
Explain 2
Investigating Arc Lengths in Concentric Circles
Consider a set of concentric circles with center O and radius 1, 2, and 3,
and so on. The central angle shown in the figure is a right angle and it
cuts off arcs that measure 90°.
90°
1
2
O
3
Example 2 Find and graph arc lengths for different radii.
 For each value of the radius
r listed in the table below, find the corresponding arc length.
Write the length in terms of π and rounded to the nearest hundredth.
For example, when
90 · 2π(1) = _
1 π ≈ 1.57.
r =1, the arc length is _
2
360
© Houghton Mifflin Harcourt Publishing Company
Radius r
1
2
3
4
5
Arc length s in
terms of π
1π
_
2
π
_3π
2π
_5π
Arc length s to the
nearest hundredth
1.57
3.14
4.71
6.28
7.85
2
 Plot the ordered pairs from your table on the coordinate plane.
2
s
8
6
4
2
r
What do you notice about the points?
The points lie on a straight line through the origin.
0
2
4
6
8
What type of relationship is the relationship between arc length and radius?
proportional relationship (direct variation)
What is the constant proportionality for this relationship?
1π
2
_
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Reflect
What happens to the arc length when you double the radius? How is this connected to
the idea that all circles are similar?
The arc length doubles. Since all circles are similar, the ratio of the radii (2:1) equals the
4.
ratio of the arc length (2:1).
Converting Radian Measure
Explain 3
As you discovered in Explain 2, when the central angle is fixed at m°, the length of the arc cut
off by a central angle is proportional to (or varies directly with) the radius. In fact, you can see
that the formula for arc length is a proportional relationship when m is fixed.
m
s = _ ⋅ 2πr
360
constant of proportionality
m ⋅ 2π. This constant of
The constant of proportionality for the proportional relationship is _
360
proportionality is defined to be the radian measure of the angle.
Example 3 Convert each angle measure to radian measure.

180°
To convert to a radian measure, let
m ⋅ 2π.
m = 180 in the expression _
360
180 ⋅ 2π Substitute 180 for m.
180° = _
360
= π radians
Simplify.

60°
To convert to a radian measure, let
__π
= 3
Substitute 60 for m.
radians Simplify.
Reflect
5.
Explain why the radian measure for an angle m° is sometimes defined as the length of
the arc cut off on a circle of radius 1 by a central angle of m°.
By the arc length formula, the length of the arc cut off on a circle of radius 1 by a central
m
angle of m° is
∙ 2π, and this is the radian measure of the angle.
360
_
6.
π.
Explain how to find the degree measure of an angle whose radian measure is_
4
π
m
m
1
Solve =
∙ 2π. Dividing both sides by π and simplifying gives =
and
4
4
360
180
multiplying both sides by 180 shows that m = 45, so the angle measures 45°.
_ _
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60
60° = _ ⋅ 2π
360
m ⋅ 2π.
m = 60 in the expression _
360
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