Area of Circular Sector

Types of sectors

There are two types of sectors formed when a circle is divided.

Major sector

This sector is the larger portion of the circle. It has a larger angle which is greater than 180 degrees.

Minor sector

The minor sector is the smaller portion of the circle. It has a smaller angle which is less than 180 degrees.

An illustration of the major and minor sectors, Njoku - StudySmarter Originals

How to calculate the area of a sector?

Deriving the area formula using the subtended angle by the sector

Using angles in degrees.

Let us remark that the angle covering the whole circle is 360 degrees, and we recall that the area of a circle is πr².

A sector is a portion of a circle containing two radii and an arc, and hence our aim is to find a way to reduce the circle until we find an arc.

Step 1.

The circle is whole, we are thus considering the angle 360 degrees, so the area is

$Are{a}_{circle}=\pi r^2$.

Step 2.

From the above diagram, the circle has been divided into half. This means that the eared of each of the obtained semicircles is,

$Are{a}_{semicircle}=\frac{1}{2}\pi r^2$.

Note that the angle subtended by the semicircle is 180 degrees which is half of the subtended angle in the center of the whole circle. By dividing 180 degrees by 360 degrees, we get that $\frac{1}{2}$ which multiplies the area of the circle. In other words,

$Are{a}_{semicircle}=\frac{180}{360}{\mathrm{\pi r}}^2=\frac{1}{2}{\mathrm{\pi r}}^2.$

Step 3.

Now we divide the semicircle to get a quarter of a circle. Hence the area of the quarter of the circle will be

$Are{a}_{quarterofthecircle}=\frac{1}{4}\pi r^2$.

Note that the angle formed by the quarter of a circle is 90 degrees, which is the quarter of the subtended angle by the whole circle. By dividing 90 degrees by 360 degrees, we get that $\frac{1}{4}$which multiplies the area of the circle. In other words,

$Are{a}_{quarterofthecircle}=\frac{90°}{360°}{\mathrm{\pi r}}^2=\frac{1}{4}{\mathrm{\pi r}}^2.$

Step 4.

The above steps can be generalized to any angle $\theta$. In fact, we can deduce that the angle subtended by the sector of a circle determines the area of that sector and so we have

$Are{a}_{sector}=\frac{\theta }{360}\pi r^2.$

where θ is the angle subtended by the sector and $r$ is the radius of the circle.

Using angles in radians.

Sometimes, rather than giving you the angle in degrees, your angle is given in radians. The are of the sector is thus,

$Are{a}_{sector}=\frac{\theta }{2}{r}^2$

How is this formula derived?

We recall that $180°=\mathrm{\pi }\mathrm{radians}$, thus$360°=2\pi$.

Now, replace in the formula for the area of the sector, derived earlier in the article, we get

$$\begin{gathered} A_{sector}=\frac{\theta }{360}\times \pi r^2 \\ Are{a}_{sector}=\frac{\theta }{2\pi }\times {\mathrm{\pi r}}^2 \\ {\mathrm{Area}}_{\mathrm{sector}}=\frac{\mathrm{\theta }}{2}{\mathrm{r}}^2. \end{gathered}$$

Using the arc length

If the length of an arc is given, you can also calculate the area of a sector.

We recall first the circumference of the circle,

$Circumferenceofacircle=2\pi r$.

Note that the arc is a part of the circumference of the circle which is determined by the subtended angle $\theta$.

Assuming that $\theta$is expressed in degrees, we have

$arclength=\frac{\theta }{360°}\times 2\pi r$.

Now recall the area formula of the arc subtended by the angle $\theta ,$

$Are{a}_{sector}=\frac{\theta }{360}\pi r^2$,

and this can be rewritten in the following

$Are{a}_{sector}=\frac{\theta }{360}\pi r^2=\frac{\theta }{360.2}\times 2\times \pi r\times r=\frac{\theta }{360}\times 2\times \pi r\times \frac{r}{2}=arclength\times \frac{r}{2}$

Thus,

$Are{a}_{sector}=arclength\times \frac{r}{2}.$