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

What Is the Area of a Sector of a Circle When the Angle Is in Radians?

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



where θ is the angle (in radians) 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 When the Angle Is in Radians

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 the angle of the sector θ.
θ × r2.

Step 3
Divide by 2.
θr22.

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 When the Angle Is in Radians

Question: What is the area of the sector with an angle of 2 radians and a radius of 5cm, as shown below?



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

Step 2
Multiply by the angle of the sector, 2 radians.
2 × 25 = 50cm2.

Step 3
Divide by 2.
50 ÷ 2 = 25cm2.

The area of the sector with an angle of 2 radians and a radius of 5cm is 25cm2.

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

The slider below shows another real example of how to find the area of a sector of a circle when the angle is in radians:

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

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

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



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



WHAT ARE RADIANS?

Radians are a way of measuring angles.

1 radian is the angle found when the radius is wrapped around the circle.



There are 2π radians in a full angle.

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 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:



The π's cancel, leaving the simpler formula:



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.