Area of a Circular Sector Formula

Geometry Formulas: Area, Volume, Perimeter
Formula:

What is the formula for the area of a circular sector?

Answer

Area of a circular sector
A = (deg / 360) · π · r²

Here r is the radius, deg is the angle in degrees. The result comes out in square units.

Where the formula comes from

A = (deg / 360) · π · r²

A sector is a slice of the disc, and the slice takes the same fraction of the area as its angle takes of the full turn. A quarter of 360° is a quarter of πr².

Worked examples

Let r = 6, deg = 90. Then the area of a circular sector ≈ 28.27 square units.

Let r = 4.5, deg = 120. Then the area of a circular sector ≈ 21.21 square units.

The common mistake

Feeding radians into a formula written for degrees. If the angle is given in radians the fraction is θ/2π, not θ/360, and the two differ by a factor of about 57.

Related Calculations

See Also

Geometry Formulas: Area, Volume, Perimeter

These are the formulas for the area, perimeter, circumference, volume and surface area of the standard shapes — 31 of them. Each page shows not only the formula but where it comes from, what every letter stands for, two worked examples and the mistake that this particular shape invites. For example, it can help you find out what is the formula for the area of a circular sector? (The answer is: A = (deg / 360) · π · r²). Pick the pair — "Area of a circle", for instance — and hit the "Find out" button.
Geometry Formulas: Area, Volume, Perimeter
Formula:

Values from the Formula (deg = 90)

If r =Area
1≈ 0.79 square units
2≈ 3.14 square units
3≈ 7.07 square units
4≈ 12.57 square units
5≈ 19.63 square units
10≈ 78.54 square units

FAQ

What is the formula for the area of a circular sector?

Area of a circular sector A = (deg / 360) · π · r²

Where does this formula come from?

A sector is a slice of the disc, and the slice takes the same fraction of the area as its angle takes of the full turn. A quarter of 360° is a quarter of πr².

What is the most common mistake here?

Feeding radians into a formula written for degrees. If the angle is given in radians the fraction is θ/2π, not θ/360, and the two differ by a factor of about 57.

How do I use this formula?

Put your own numbers in place of the letters. With r = 4.5, deg = 120, for instance, the formula A = (deg / 360) · π · r² gives ≈ 21.21 square units.

See Also