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

The Lesson

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

area equals half time radius squared times angle In this formula, r is the radius of the circle and θ is the angle (in radians) of the sector. The image below shows what we mean by the area of a sector:

angle, radius and sector

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

Finding the area of a sector of a circle, when the angle is in radians, is easy.

Question

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

Step-by-Step:

1

Start with the formula:
Area of sector = 12 r2θ
Don't forget: r2 = r × r (r squared).

2

Substitute the angle and the radius into the formula. In our example, θ = 2 and r = 5.
Area of sector = 12 × 52 × 2 Area of sector = 12 × 5 × 5 × 2 Area of sector = 25 cm2

Answer:

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

Lesson Slides

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. 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

What Are Radians?

Radians are a way of measuring angles. 1 radian is the angle found when the radius is wrapped around the circle.

1 radian equals 1 radius wrapped around circumference

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 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:
Area of sector = θ × πr2
The πs cancel, leaving the simpler formula:
Area of sector = θ2 × r2 = 12 r2θ

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.
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See Also

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