I want to talk about how to use the definition of radian measure to calculate the measure of an angle. Now recall the definition of a radian measure assumes that an angle has been drawn with its vertex in center of a circle and the measure of the angle is theta.
Now if r is the radius of the circle and s is the length of the arc intercepted, then theta equals s over r and that's the definition of the radian measure, the arc length divided by the radius. Let's try this with an example.
Problem; find the measure of the angle in the picture. So we have an angle theta, it intercepts an arc whose length is 20 inches and the radius of the circle is 8 inches, so theta equals the arc length 20 inches divided by the radius 8 inches and we see inches are going to cancel and a lot of other things will cancel too. 20 is 4 times 5, 8 is 4 times 2, so this is 5 halves and we say radians. So the measure of the angle is 2 and a half radians.