# Understanding Radian Measure

A circle O with radius 1 unit has its center placed at the origin. Let *A* be its intersection with the *x*-axis at (1,0) and* P* be another point on its circumference. If we move *P* along its circumference, then we can determine the distance traveled by *P*. If we let* A* be the starting point of* P* as it moves counterclockwise, then the distance traveled by *P* is equal to the length of arc *AP* represented by the red arc in the following figure.

To be able to know the length of arc *AP*, first, we must know the total distance traveled by *P *from *A* going counterclockwise and back to *A* (i.e. complete revolution). That is, we need to find the circumference of the circle. Since a unit circle has radius 1 unit, its circumference *C* is

.

We know that as *P* moves, we can also measure angle *AOP* in terms of degrees. Now, one complete revolution is equal to 360 degrees. Therefore, degrees and degrees.

To find the amount of rotation of *P* in terms of degrees after traveling 1 unit, we have

This gives us the following equations,

.

We know that , so we get degrees. This means that the position of *P* where the arc length of *AP* is equal to 1 is the same as the position of *P* when it is rotated approximately 57.29 degrees counterclockwise about the center of the circle from *A*. From here, it is not difficult to observe that for each amount of rotation in degrees, there is also a corresponding arc length. Therefore, we can also consider arc length as a unit for amount of rotation. This unit is called **radian**.

In addition, instead of calculating for approximated values (such as 57.29), we can just leave the exact values. For example, at (-1,0), the point is rotated half the distance on the circumference so it is radians. Halfway between (1,0) and is radians. At (0,-1) which is radians.