Definitions:
A radian is the measure of the central angle subtended by an arc equal in length to the radius of a circle. The SI symbol for a radian is $rad$
The circular measure of an angle is the number of radians it contains.
Question body:
Consider the formula below which is used to calculate the area of a sector of a circle:
$$s = \frac{1}{2}\theta r^{2}$$
In many textbooks for introductory geometry, authors emphasize that the formula holds if $\theta$ is given in radians. However, when we compute, we use the circular measure of an angle given in radians. For example, let the radius of a circle be $6$ and the angle subtended at the centre be $\frac{\pi}{3}$ $rad$, hence we compute as follows:
$$s = (\frac{1}{2})(\frac{\pi}{3})(6^{2}) = 6\pi$$
As you can see, $rad$ is never included because we consider the circular measure of an angle, which is given in radians. Therefore, it yields $6\pi$ and not $6\pi$ $rad$.
Question:
As the title suggests, does $\theta$ represent the cirular measure of an angle or just an angle given in radians? Also, please explain why.
Best Answer
Yes. The angle must be given in radians, or it can be the circular measure, or arc length, of the arc on the unit circle (a circle with radius 1). Think about a circle whose area is $s=\pi r^2$, and we want a portion of that area. Since a circle contains $2\pi$ radians, the area of the sector would be $s=\pi r^2\cdot\frac{\theta \textrm{ rad}}{2\pi\textrm{ rad}}$ and notice that it has to be radians here. Thus, $s=\frac{1}{2}\theta r^2$.
Also, a similar formula can be derived with degrees, but the denominator should be $360^\circ$ instead of $2\pi$. Thus, $s=\frac{\theta^\circ}{360^\circ}\pi r^2$, and remember that $\theta$ here is in radians.