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

Thus $$m\angle AOB = 1$$ radian.

Relationship between the length of an arc of a circle and the circular measure of its central angle:

Prove that $$\theta = \frac{l}{r}$$
Where $$r$$ is the radius of the circle, $$l$$ is the length of the arc and $$\theta $$ is the circular measure of the central angle.

Proof:

Let there be a circle with center $$O$$ and radius $$r$$. Suppose that the length of the arc and the central angle are $$m\angle AOB = \theta $$ radian. Take an arc of length of $$ = r$$.

By definition, $$m\angle AOC = 1$$ 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 $$\theta $$ (in radian) subtended by a circular arc of length $$l$$ is given by $$\theta = \frac{l}{r}$$, where $$r$$ is the radius of the circle.

Remember that $$r$$ and $$l$$ 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 $$r = 3\,cm$$ and $$l = 6\,cm$$

Then
\[\theta = \frac{l}{r}\,\,\,\, = \,\,\,\,\frac{6}{3} = 2\]