Radian measure: A measurement of angles based on a ratio of lengths.

Angles are often measured or specified in degrees, but another unit, the radian, is useful in many computations. The radian measure of an angle is the ratio of two lengths on a circle. The angle and lengths are perhaps most easily understood by looking at the diagram in Equation 1 on the right. The arc length is the length of the arc on the circumference cut off by the angle when it is placed at the circle’s center. The other length is the radius of the circle. The radian measure of the angle equals the arc length divided by the radius.

A 360° angle equals 2π radians. Why is this so? The angle 360° describes an entire circle. The arc length in this case equals the circumference (2πr) of a circle divided by the radius r of the circle. The radius factor cancels out, leaving 2π as the result.

Radians are dimensionless numbers. Why? Since a radian is a ratio of two lengths, the length units cancel out. However, we follow a radian measure with "rad" so it is clear what is meant.