Area of a Sector of a Circle
(KS3, Year 7)

The Lesson

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

area of sector equals angle divided by 360 all times pi times radius squared In this formula, 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:

area, angle and radius

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?
area of sector with angle of 72 degrees and a radius of 5 cm

Step-by-Step:

1

Start with the formula:
Area of sector = θ360° × πr2
Don't forget: π is pi (≈ 3.14), / means ÷ and r2 = r × r (r squared).

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 cm2

Answer:

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

Lesson Slides

The slider below shows another real example of how to find the area of a sector of a circle. Open the slider in a new tab

What Is a Sector?

A sector is a region of a circle bounded by two radii and the arc lying between the radii.

sector

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 πr2, where r is the radius. For example, a sector that is half of a circle is half of the area of a circle.

half a circle area A sector that is quarter of a circle has a quarter of the area of a circle.

quarter of a circle area In each case, the fraction is the angle of the sector divided by the full angle of the circle.

angle in a sector 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 = θ360° × πr2

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.
Help Us To Improve Mathematics Monster

  • Do you disagree with something on this page?
  • Did you spot a typo?
Please tell us using this form

See Also

What is a circle? What is an angle? What is pi? What are degrees? Finding the area of a circle What is a full angle?