# Circular System

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**:

A 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 a circle whose rays cut off an arc on a circle whose length is equal to the radius .

Thus radian.

__Relationship 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 the length of the arc and the central angle are radian. Take an arc of length of .

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