AREA OF A SECTOR

A sector of a circle is a pie-shaped region bounded by an arc and an angle.
On this page we'll show you how to find the arc length a and the area A of the sector.


In both calculations we'll need to know the radius r of the circle, and the central angle of the sector.

Finding Arc Length

We'll use a circle with radius 15 cm and central angle 60°.

The first step is to find out what fraction of the circle
is represented by the central angle 60°:

The sector represents 1/6 of the circle,
or 0.1666... of the circle.

To find the arc length a, we now need to find the circumference (the distance around the full circle), and then find 1/6 of it (multiply it by 0.1666...)

This is the length of the arc a.
This method will work for any sector in any circle. You just need to find the fraction of the circle that the sector angle represents, and then find that fraction of the circumference.



Here's a quicker formula to use, that summarizes what we did in one calculation.

Note: Math 30 students will learn a new formula for finding arc length: a = r , where is measured in radians.


Finding the Sector Area


Again we'll use a circle with radius 15 cm and central angle 60°.

The first step once again is to find out what fraction of the circle
is represented by the central angle 60°:

The sector represents 1/6 of the circle,
or 0.1666... of the circle.

To find the sector area A, we now need to find the area of the full circle, and then find 1/6 of it (multiply it by 0.1666...)

This is the area of the sector, A.
This method will work for any sector in any circle. You just need to find the fraction of the circle that the sector angle represents, and then find that fraction of the area.



Here's a quicker formula to use, that summarizes what we did in one calculation.