Radians and arc length

Radians and arc lengthAn alternate definition of radians is sometimes given as a ratio. Instead of taking the unit circle with center at the vertex of the angle, take any circle with center at the vertex of the angle. Then the radian measure of the angle is the ratio of the length of the subtended arc to the radius of the circle. For instance, if the length of the arc is 3 and the radius of the circle is 2, then the radian measure is 1.5.
The reason that this definition works is that the length of the subtended arc is proportional to the radius of the circle. In particular, the definition in terms of a ratio gives the same figure as that given above using the unit circle. This alternate definition is more useful, however, since you can use it to relate lengths of arcs to angles. The formula for this relation is
radian measure times radius = arc length
For instance, an arc of 0.3 radians in a circle of radius 4 has length 0.3 times 4, that is, 1.2.