sector of a circle

A sector is a fraction of the interior of a circle, described by a central angle $\theta$.If $\theta =2\pi ,$ the sector becomes a complete circle.

If the central angle is $\theta ,$ and the radius of the circle is $r,$ then the area of the sector is given by

$$\frac{1}{2}{r}^2\theta$$

This is obvious from the fact that the area of a sector is $\frac{\theta }{2\pi }$ times the area of the circle (which is $\pi r^2$).Note that, in the formula, $\theta$ is in radians.

Remark. Since the length $a$ of the arc of the sector is $r\theta$, the area of the sector is $\frac{1}{2}ar$, which is equal to the area of a triangle with base $=a$ and the height $=r$.