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How to Find the Area of a Sector of a Circle (Mathematics Lesson)

What Is the Area of a Sector of a Circle?

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



where θ is the angle (in degrees) subtended at the center of the sector and r is the radius of the circle.

How to Find the Area of a Sector of a Circle

Question: What is the area of the sector with angle θ and radius r, as shown below?



Step 1
Multiply the radius r by itself.
r × r = r2.

Step 2
Multiply by pi (symbol: π)
π × r2 = πr2.

Step 3
Multiply by the angle of the sector θ.
θ × πr2.

Step 4
Divide by 360°.
θ360° × πr2.

This is the area of the sector with angle θ and radius r.

A Real Example of How to Find the Area of a Sector of a Circle

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



Step 1
Multiply the radius, 5cm, by itself.
5 × 5 = 25cm2.

Step 2
Multiply by pi (symbol: π).
π × 25 = 3.14 × 25 = 78.5cm2.

Step 3
Multiply by the angle of the sector, 72°
72° × 78.5 = 5,654cm2.

Step 4
Divide by 360°.
5,654 ÷ 360° = 15.7cm2.

The area of the sector with an angle of 72° and a radius of 5cm is 15.7cm2.

Another Real Example of How to Find the Area of a Sector of a Circle

The slider below shows another real example of how to find the area of a sector of a circle:

How to Find the Area of a Sector of a Circle When the Angle Is in Radians

The formula used on this page works when the angle subtended at the center of the sector is in degrees.

When the angle subtended at the center of the sector is given in radians, the formula for the area of the sector becomes:



Read more about how to find the area of a sector (in radians)
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Note
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.

For example, a sector that is half of a circle has 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:



IS THE ANGLE GIVEN IN DEGREES OR RADIANS?

The formula to find the area 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.