1. 1.2 Radian Measure, Arc Length, and Area
Another way to measure angles is using what is called radians.
Given a circle of radius r with the vertex of an angle as the center of the circle, if the arc length formed by intercepting the circle with the sides of the angle is the same length as the radius r, the angle measures one radian.
initial side
terminal side
radius of circle is r r
arc length is also r
This angle measures 1 radian
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2. 1.2 Radian Measure, Arc Length, and Area
We need a conversion from degrees to radians. We could use a conversion fraction if we knew how many degrees equaled how many radians.
180 º =  radians
Convertion: degrees radians
1. Change to degrees
a. 4/3 b. /8
Change to radians
a º b. 54º
3. Express  = 4 in terms of degrees, minutes, seconds
(Change 4 radians to degrees, minutes, and seconds.)
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3. A Sense of Angle Sizes
See if you can guess the size of these angles first in degrees and then in radians.
You will be working so much with these angles, you should know them in both degrees and radians.
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4. 1.2 Radian Measure, Arc Length, and Area
Arc length s of a circle is found with the following formula:
arc length
radius
measure of angle
IMPORTANT: ANGLE MEASURE MUST BE IN RADIANS TO USE FORMULA!
s = r
Figure 1.17
Find the radius of a circle in which an arc of 3 km subtends a central angle of 20°.
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5. 1.2 Radian Measure, Arc Length, and Area
Find the arc length if we have a circle with a radius of 3 meters and central angle of 0.52 radian. 3
 = 0.52
arc length to find is in black
s = r
0.52
= 1.56 m
If the measure of the angle is in degrees, we can't use the formula until we convert it to radians.
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6. 1.2 Radian Measure, Arc Length, and Area
Area of a Sector of a Circle
The formula for the area of a sector of a circle (shown in red here) is derived in your textbook. It is:  r
Again  must be in RADIANS so if it is in degrees you must convert to radians to use the formula.
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7. Given an arc of length 4 ft and a circle of radius 7 ft,
find the exact radian measure of the central angle
subtended by the arc; then find the area of the sector
determined by the central angle.
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