Area of a Sector

1. What is a Sector?

A sector is a part of a circle enclosed by two radii and the arc between them. It looks like a slice of a pizza or a piece of a pie.

The size of a sector depends on the central angle \((\theta)\) and the radius \((r)\).

The area of a sector is a fraction of the area of the whole circle.

If the central angle is \(\theta\) degrees, then:

A_{\text{sector}} = \dfrac{\theta}{360^\circ} \times \pi r^2

If the central angle is in radians:

A_{\text{sector}} = \dfrac{1}{2} r^2 \theta

The whole circle has an angle of \(360^\circ\) and area \(\pi r^2\). A sector with angle \(\theta\) is just a fraction of the full circle.

The fraction is:

\dfrac{\theta}{360^\circ}

Multiplying this with the area of the full circle gives the sector area.

Both follow the same formula for area.

Here are clear examples to understand sector area calculation:

In a circle of radius \(r = 6\text{ cm}\), find the area of a sector with angle \(\theta = 90^\circ\).

A = \dfrac{90}{360} \times \pi \times 6^2 = \dfrac{1}{4} \times 36\pi = 9\pi\text{ cm}^2

Find the area of a sector of radius \(r = 5\text{ cm}\) and angle \(\theta = \dfrac{\pi}{3}\) radians.

A = \dfrac{1}{2} r^2 \theta = \dfrac{1}{2} \times 25 \times \dfrac{\pi}{3} = \dfrac{25\pi}{6}\text{ cm}^2

Sector area is expressed in square units such as:

Sector area is used in: