# Area of a Sector of a Circle

(KS3, Year 7)

**θ**is the angle (in degrees) of the sector and

**r**is the radius of the circle. The image below shows what we mean by the area of a sector:

## How to Find the Area of a Sector of a Circle

Finding the area of a sector of a circle is easy.## Question

What is the area of the sector with an angle of 72° and a radius of 5 cm, as shown below?## Step-by-Step:

## 1

## 2

Substitute the angle and the radius into the formula. In our example, θ = 72° and r = 5.

Area of sector = ^{72°}⁄_{360°} × π × 5 × 5

Area of sector = (72° ÷ 360°) × 25 × π

Area of sector = 15.7 cm^{2}

## Answer:

The area of a sector of a circle with a radius of 5 cm, with an angle of 72°, is 15.7 cm^{2}.

## What Is a Sector?

A sector is a region of a circle bounded by two radii and the arc lying between the radii.## Why Does the Formula Work?

The area of a sector is just a fraction of the area of the circle of the same radius. The area is given by**πr**, where

^{2}**r**is the radius. For example, a sector that is half of a circle is half of the area of a circle. A sector that is quarter of a circle has a quarter of the area of a circle. In each case, the fraction is the angle of the sector divided by the full angle of the circle. When measured in degrees, the full angle is 360°. Hence for a general angle θ, the formula is the fraction of the angle θ over the full angle 360° multiplied by the area of the circle:

Area of sector =

^{θ}⁄_{360°}× πr^{2}## Beware

## Is the Angle Given in Degrees or Radians

The formula to find the length of a sector of a circle depends on whether the angle at the center of the sector is given in degrees or radians. Make sure you check what units the angle is given in.## Worksheet

This test is printable and sendable