Question

Define radian measure of an angle.

Answer 1

The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. One way to measure angles is in radians.

In order for us to be able to define radians, it is necessary to introduce the concept of a central angle. A central angle is an angle whose vertex is at the center of a circle. In the circle below, the center is point $O$, the length of the radius is $r,$ and $\angle AOB$ is a central angle.

Notice that $\angle AOB$ cuts off or determines an arc $AB$ that has length $s$. The radian measure of a central angle, often denoted by the Greek letter theta $\left(\theta \right)$, is defined to be the ratio of the arc length to the length of the radius. So the radian measure of is given by:

$\theta =\frac{arclength}{radiuslength}=\frac{s}{r}$

arc length $s$ and radius length $r$ must be in same units