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# Area of Sector – Explanation & Examples

To recall, **a sector is** a portion of a circle enclosed between its two radii and the arc adjoining them.

**For example**, a pizza slice is an example of a sector representing a fraction of the pizza. There are two types of sectors, minor and major sector. A minor sector is less than a semi-circle sector, whereas a major sector is a sector that is greater than a semi-circle.

**In this article, you will learn:**

- What the area of a sector is.
- How to find the area of a sector; and
- The formula for the area of a sector.

## What is the Area of a Sector?

**The area of a sector is the region enclosed by the two radii of a circle and the arc.** In simple words, the area of a sector is a fraction of the area of the circle.

## How to Find the Area of a Sector?

*To calculate the area of a sector, you need to know the following two parameters:*

- The length of the circle’s radius.
- The measure of the central angle or the length of the arc. The central angle is the angle subtended by an arc of a sector at the center of a circle. The central angle can be given in degrees or radians.

With the above two parameters, finding the area of a circle is as easy as ABCD. It is just a matter of plugging in the values in the area of the sector formula given below.

### Formula for area of a sector

There are three formulas for calculating the area of a sector. Each of these formulas is applied depending on the type of information given about the sector.

#### Area of a sector when the central angle is given in degrees

If the angle of the sector is given in degrees, then the formula for the area of a sector is given by,

**Area of a sector = (θ/360) π r^{2}**

A = **(θ/360) π r^{2}**

Where θ = the central angle in degrees

Pi (π) = 3.14 and r = the radius of a sector.

#### Area of a sector given the central angle in radians

If the central angle is given in radians, then the formula for calculating the area of a sector is;

**Area of a sector = (θ r^{2})/2**

Where θ = the measure of the central angle given in radians.

#### Area of a sector given the arc length

Given the length of the arc, the area of a sector is given by,

**Area of a sector = rL/2**

Where r = radius of the circle.

L = arc length.

Let’s work out a couple of example problems involving the area of a sector.

*Example 1*

Calculate the area of the sector shown below.

__Solution__

Area of a sector = (θ/360) πr^{2}

= (130/360) x 3.14 x 28 x 28

= 888.97 cm^{2}

*Example 2*

Calculate the area of a sector with a radius of 10 yards and an angle of 90 degrees.

__Solution__

Area of a sector = (θ/360) πr^{2}

A = (90/360) x 3.14 x 10 x 10

= 78.5 sq. yards.

*Example 3*

Find the radius of a semi-circle with an area of 24 inches squared.

__Solution__

A semi-circle is the same as half a circle; therefore, the angle θ = 180 degrees.

A= (θ/360) πr^{2}

24 = (180/360) x 3.14 x r^{2}

24 = 1.57r^{2}

Divide both sides by 1.57.

15.287 = r^{2}

Find the square root of both sides.

r = 3.91

So, the radius of the semi-circle is 3.91 inches.

*Example 4*

Find the central angle of a sector whose radius is 56 cm and the area is 144 cm^{2}.

__Solution__

A= (θ/360) πr^{2}

144 = (θ/360) x 3.14 x 56 x 56.

144 = 27.353 θ

Divide both sides by θ.

θ = 5.26

Thus, the central angle is 5.26 degrees.

*Example 5*

Find the area of a sector with a radius of 8 m and a central angle of 0.52 radians.

__Solution__

Here, the central angle is in radians, so we have,

Area of a sector = (θ*r*^{2})/2

= (0.52 x 8^{2})/2

= 16.64 m^{2}

*Example 6*

The area of a sector is 625mm^{2}. If the sector’s radius is 18 mm, find the central angle of the sector in radians.

__Solution__

Area of a sector = (θ*r*^{2})/2

625 = 18 x 18 x θ/2

625 = 162 θ

Divide both sides by 162.

θ = 3.86 radians.

*Example 7*

Find the radius of a sector whose area is 47 meters squared and central angle is 0.63 radians.

__Solution__

Area of a sector = (θ*r*^{2})/2

47 = 0.63r^{2}/2

Multiply both sides by 2.

94 = 0.63 r^{2}

Divide both sides by 0.63.

r^{2} =149.2

r = 12.22

So, the radius of the sector is 12.22 meters.

*Example 8*

The length of an arc is 64 cm. Find the area of the sector formed by the arc if the circle’s radius is 13 cm.

__Solution__

Area of a sector = rL/2

= 64 x 13/2

= 416 cm^{2}.

*Example 9*

Find the area of a sector whose arc is 8 inches and radius is 5 inches.

__Solution__

Area of a sector = rL/2

= 5 x 8/2

= 40/2

= 20 inches squared.

*Example 10*

Find the angle of a sector whose arc length is 22 cm and the area is 44 cm^{2}.

__Solution__

Area of a sector = rL/2

44 = 22r/2

88 = 22r

r = 4

Hence, the radius of the sector is 4 cm.

Now calculate the central angle of the sector.

Area of a sector = (θ*r*^{2})/2

44 = (θ x 4 x 4)/2

44 = 8 θ

θ =5.5 radians.

Therefore, the central angle of the sector is 5.5 radians.