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.