Sector of a Circle

A sector of a circle is a portion enclosed between two radii and the arc connecting them. The perfect and simplest example of this is a slice of pizza or a pie. Sectors play a crucial role in geometry as their applications are used in engineering, architecture, and even graphic design. 

In this guide, you will learn about sectors, their types, and how to calculate their area. Let’s dive in!

1.0Types of Sectors in a Circle

The types of sectors depend on the sector angle. Sector angle is the angle subtended at the centre of the circle by the arc. They are of two types:

A semicircle is a type of sector with an angle equal to 180°. Knowing the type of sector is important for understanding symmetry and calculating the area of the sector.

2.0How to Find the Area of a Sector?

To calculate the area of sector, you need two things: the radius of the circle and the sector angle. There are two formulas you can use for this:

Area of Sector Formula (Angle in Degrees)

Area = θ/360 x πr²

Where:

• θ = Sector angle in degrees
• r = Radius of the circle
• Π = 3.1416

Area of Sector Formula (Angle in Radians)

Area = 12r²θ

Where:

• θ = Sector angle in radians
• r = Radius

3.0Arc Length and Perimeter 

There are other properties of a sector that you might need to calculate. Here are the specific formulas you can use:

Arc length

Arc length = (θ/360) x 2πr (This formula is for degrees)

Arc Length = rθ (This formula is for radians)

Perimeter

Perimeter = 2r + Arc length

4.0Key Terms

Let’s take a look at the structure of a sector:

5.0Solved Problems

Example 1: Area of a Sector (Degrees)

Problem:

Find the area of a sector with a radius of 7 cm and a central angle of 60°.

Solution:

Area = (60/360) x Π x 7²

= (⅙) x Π x 49

Area = 25.66 cm²

Example 2: Arc Length (Radians)

Problem:

If the radius is 10 cm and the sector angle is Π/3 radians, find the arc length.

Solution:

Arc length = r = 10 x Π/3 = 10.47 cm

Example 3: Perimeter of a Sector

Problem:

Radius = 4 cm, Angle = 90°. Find the perimeter of the sector.

Solution:

First, find the arc length:

Arc length = (90/360) x 2Π x r

= (¼) x 8Π

= 2Π

Arc length = 6.28 cm

Perimeter = 2r + arc length

= 2 x 4 + 6.28

Perimeter = 14.28 cm

Example 4: Area of Sector (Radians)

Problem:

Radius = 5 cm, Sector angle = 2 radians. Find the area.

Solution:

Area = 12r²

=(½) x 5² x 2

=(½) x 25 x 2

= 25 cm²