# Area of Circular Sector

Who doesn't love pizza? When next you get a pizza delivery, as it is being shared with your friend and family look closely at each piece, you've got a sector not just pizza! Herein, you shall have a better look into the size of each piece of pizza (sector).

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## What is a sector?

A sector is a portion of a circle bounded by two radii and an arc. A typical sector can be seen when a pizza is shared in 8 portions for instance. Every portion is a sector taken from the circular pizza. A sector also subtends an angle where its two radii meet. This angle is very important because it tells us what proportion of the circle is occupied by the sector.

A diagram illustrating the sector of a circle, Njoku - StudySmarter Originals

### 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.

The area of a sector subtended by an angle $\theta$ (expressed in degrees) is given by

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

Calculate the area of a sector with angle 60 degrees at the center and having a radius of 8cm. Take $\pi =3.14.$

Solution.

First, we define our variables, $\theta =60°,r=8cm$.

The area of the sector is given by,

${A}_{sector}=\frac{\theta }{360°}\pi {r}^{2}\phantom{\rule{0ex}{0ex}}Are{a}_{sector}=\frac{60°}{360°}×3.14×{8}^{2}\phantom{\rule{0ex}{0ex}}Are{a}_{sector}=\frac{1}{6}×3.14×64\phantom{\rule{0ex}{0ex}}Are{a}_{sector}=33.49c{m}^{2}.\phantom{\rule{0ex}{0ex}}$

Thus the area of the sector subtended by an angle of 60 degrees in a circle of radius 8 cm is 33.49 cm squared. " height="19" id="2366077" role="math" style="max-width: none;" width="9"> $c{m}^{2}$

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

${A}_{sector}=\frac{\theta }{360}×\pi {r}^{2}\phantom{\rule{0ex}{0ex}}Are{a}_{sector}=\frac{\theta }{2\pi }×{\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}{\mathrm{Area}}_{\mathrm{sector}}=\frac{\mathrm{\theta }}{2}{\mathrm{r}}^{2}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

The area of a sector subtended by an angle $\theta$ (expressed in radians) is given by

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

Calculate the area of a sector with diameter 2.8 meters with a subtended angle of 0.54 radians.

Solution.

We define our variables, r = 2.8m, θ = 0.54 radians.

The area of the sector is given by

$Are{a}_{sector}=\frac{\theta }{2}{r}^{2}.\phantom{\rule{0ex}{0ex}}Are{a}_{sector}=\frac{0.54}{2}×2.{8}^{2}\phantom{\rule{0ex}{0ex}}Are{a}_{sector}=0.27×7.84\phantom{\rule{0ex}{0ex}}Are{a}_{sector}=2.12{m}^{2}$

#### 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

$arclength=\frac{\theta }{360°}×2\pi r\phantom{\rule{0ex}{0ex}}$.

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

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

Thus,

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

The above calculation can also be done if the subtended angle is measured in radians.

The area of a sector subtended by an angle $\theta$, given its arc length is given by $Are{a}_{sector}=arclength×\frac{r}{2}.$

Find the area of a sector with arc length 12cm and radius 8cm.

Solution.

We define our variables, r = 8cm, arc length = 12cm.

The area of the sector is given by

$Are{a}_{sector}=Arclength×\frac{r}{2}\phantom{\rule{0ex}{0ex}}Are{a}_{sector}=12×\frac{8}{2}\phantom{\rule{0ex}{0ex}}Are{a}_{sector}=12×4\phantom{\rule{0ex}{0ex}}Are{a}_{sector}=48c{m}^{2}.$

## Area of Circular Sectors - Key takeaways

• A sector is a portion of a circle bounded by two radii and an arc.
• The major and minor sectors are two types of sectors formed when a circle is divided.
• The area of a sector subtended by an angle $\theta$ can be calculated through t5he information given on that angle or through its arc length.

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How do you find the area of circular sector?

You can find the area of a circular sector by multiplying the area of a circle by the angle divided by 360 degrees.

How do you derive the area of circular sector?

To derive the area of a sector, the area of a complete circle must be considered. Then the circle is reduced to its semicircle and afterwards to its quarter-circle. The application of proportion on the area of a circle considering the angle subtended by each circle ratio shows us how the area of a sector is arrived at.

What is an example of area of circular sector ?

An example of an area of a circular sector is when an angle is given with the radius of the sector and you are asked to calculate the area of the sector.

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