There is another system of angular measurement, called the Circular System. It is most useful for the study of higher mathematics. Especially in Calculus, angles are measured in radians.

** Radian**:

Radian is the measure of the angle subtended at the center of the circle by an arc, whose length is equal to the radius of the circle.

Consider a circle of radius. Construct an angle at the center of the circle whose rays cut off an arc on the circle whose length is equal to the radius .

Thus radian

__Relation between the length of an arc of a circle and the circular measure of its central angle__**:**

Prove that

Where is the radius of the circle , is the length of the arc and is the circular measure of the central angle.

__Proof__**:**

Let there be a circle with center and radius . Suppose that length of arc and the central angle radian. Take an arc of length .

By definition radian.

We know from elementary geometry that measures of central angles of the arcs of a circle are proportional to the lengths of their arcs.

Thus the central angle (in radian) subtended by a circular arc of length is given by , where is the radius of the circle.

**Remember** that and are measured in terms of the same unit and the radian measure is unit-less, i.e., it is a real number.

For example, if and

Then