## The Lesson

The area of a sector of a circle is given by the formula:In this formula,

**r**is the radius of the circle and

**θ**is the angle (in radians) of the sector. The image below shows what we mean by the area of a sector:

## How to Find the Area of a Sector of a Circle (Radians)

Finding the area of a sector of a circle, when the angle is in radians, is easy.## Question

What is the area of the sector with an angle of 2 radians 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 = 2 and r = 5.

Area of sector =

^{1}⁄_{2}× 5^{2}× 2 Area of sector =^{1}⁄_{2}× 5 × 5 × 2 Area of sector = 25 cm^{2}## Answer:

The area of a sector of a circle with a radius of 5 cm, with an angle of 2 radians, is 25 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.## What Are Radians?

Radians are a way of measuring angles. 1 radian is the angle found when the radius is wrapped around the circle.## 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 radians, the full angle is 2π. Hence for a general angle θ, the formula is the fraction of the angle θ over the full angle 2π multiplied by the area of the circle:

Area of sector =

The πs cancel, leaving the simpler formula:
^{θ}⁄_{2π}× πr^{2}
Area of sector =

^{θ}⁄_{2}× r^{2}=^{1}⁄_{2}r^{2}θ