Radian Measure

• A central angle has a measure of 1 radian if it is subtended by an arc whose length is equal to the radius of the circle.

• Consider the circle at the right. Let the radius be given by r.

• Measure an arc on the circle of length r, the radius of the circle.

• Draw a line segment from the center of the circle to point A, and name the central angle θ.

• Angle θ has a measure of 1 radian.

• Now consider a central angle that subtends an arc that is twice the length of the radius.

• Angle α has a measure of 2 radians.

• In general, if a central angle θ subtends an arc of length s, then the radian measure of θ is given by …

s

r

Example 1:A central angle θ subtends an arc of length 27cm. in a circle of radius 6cm. Determine the measure of θ.

s

r 27

6

cm

cm

9

2

4.5 radians

• Notes:

1. Radians is sometimes abbreviated rad.

2. Sometimes neither radians nor rad is added. In the previous example, we could say that the measure of the angle was 4.5, and the intent would be 4.5 radians.

3. If the measure is intended to mean degrees, then the degree symbol will be added.