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Area of a Sector of a Circle

(KS3, Year 7)

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 = ^θ⁄_360° × 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.

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 = ^θ⁄_360° × 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.