How to find the area of a sector

A sector is a section or part of a circle - shaped like a pie slice. It is enclosed by the radiuses (radii) of the circle and an arc (where the sector meets the circumference). 

The area of a sector can be easily found with a formula which uses the radius of the circle and the sector's central angle:

Area of a Sector

The central angle is located at the sector's vertex, this is the point where the sector meets the center point of the circle.

The radius (r) is the distance from the center point to the edge of the circle.

The first part of the formula is π - this is the total area of the circle.

In the next part of the formula we multiply the total area of the circle by central angle/360° - this is the fraction of the circle that the sector covers.

So, in short, to get the area of a sector, you multiply the total area of the circle by the fraction that the sector covers. See how in the worked examples below.

If you already know the fraction of the circle that the sector covers, you can simply multiply the area of the circle by this. For example, if we have a sector that covers exactly a quarter of the circle (a 90° angle), then you can multiply the area of the circle by ¼ (πr² x ¼) to get the sector area.

Worked Examples


What are the areas of the following sectors in the diagram?


i)
ADB

ii) BDC

iii) ADC



Answers

i)

Radius = 7 cm

Central angle = 72°

Substituting the radius and central angle into the sector area formula:

ADB = (π x 7²) x (72/360)

= 153.94 cm²(area of the circle) x 0.2(1/5 of the circle)

ADB = 30.79 cm²

ii)

Radius = 7cm
Central angle = 90°

We already know the area of the circle (π x 7²) from the previous question.

Circle area = 153.94 cm²

90° is exactly 1/4 of 360° this tells us that the sector covers exactly 1/4 of the circle:

BDC = 153.94cm² x 0.25(1/4) = 38.485 cm²

iii)

Radius = 7cm
Central angle = 72° + 90° = 162°

153.94 x 162/360

ADC = 153.94 x 0.45 = 69.27 cm²


Major and Minor Sectors

When there are two sectors in a circle the smaller of the two is called the minor sector, it has a central angle less than 180° and covers less than half the area of the circle.

The major sector is the larger sector and has a central angle greater than 180°, it covers more than half of the area of the circle.

Major & Minor Sectors


Sector Types

A semicircle is a sector which covers exactly half of a circle and has a central angle of 180°. It is enclosed by a semicircle arc and the diameter of the circle.

Sector Types

- An octant is a sector which covers 1/8 of a circle and has a central angle of 45°.

- A sectant is a sector which covers 1/6 of a circle and has a central angle of 60°.

- A quadrant is a sector which covers 1/4 of a circle and has a central angle of 90°.


Back to top