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 $=r$ .