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.
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 .
Relationship between the length of an arc of a circle and the circular measure of its central angle:
Where is the radius of the circle, is the length of the arc and is the circular measure of the central angle.
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