How can the radian measure of an angle determine the arc length on the unit circle?

A unit circle is the one which has radius equal to 1.

The relation between radius of circle (r), the arc length (s) and the radian measure of angle made by arc (Ф) is given as:

s = rФ

For a unit circle, r = 1, so the above equation becomes:

s= Ф

This means, the arc length will be equal to the angle made by that arc in radians.