Sector of a Circle

The area of a sector of a circle is calculated by multiplying the circle's area by the ratio of the measure of the central angle to $360^{\circ }.$

From the fact that $2\pi \text{ rad}$ equals $360^{\circ },$ an equivalent formula can be written if the central angle is given in radians.

Since the measure of an arc is equal to the measure of its central angle, the arc $AB$ measures $\theta .$ Therefore, by substituting $m\overgroup{AB}$ for $\theta ,$ another version of the formula is obtained which can also be written in degrees or radians.

Proof

Consider sector $ACB$ bounded by $\overline{AC},\overline{BC},$ and $\overgroup{AB}.$

Since a circle measures $360^{\circ },$ this sector represents $\frac{\theta }{360^{\circ }}$ of $\odot C.$ Therefore, the ratio of the area of a sector to the area of the whole circle is proportional to $\frac{\theta }{360^{\circ }}.$

$$\begin{array}{r}\frac{\text{Area of Sector}}{\text{Area of Circle}}=\frac{\theta }{360^{\circ }}\end{array}$$

Recall that the area of a circle is $\pi r^2.$ By substituting it into the equation and solving for the area of a sector, the desired formula can be obtained.

$$\begin{array}{r}\frac{\text{Area of Sector}}{\pi r^2}=\frac{\theta }{360^{\circ }}\end{array}$$

Therefore, the area of a sector of a circle can be found by using the following formula.

$\text{Area of Sector}=\frac{\theta }{360^{\circ }}\cdot \pi r^2$