Sector of a Circle

We have already studied simple plane figures and how to find their areas and perimeters. In our daily life, we come across many objects which are circular in shape, for example, wheels, bangles, circular paths, circular chowk, etc. Finding areas and perimeter of these circular objects becomes a need. 

• A circle is the collection of all the points in a plane, which are at a fixed distance from a fixed point. The fixed point is called the center of the circle and the fixed distance from a point on the circle is called the radius of the circle.

Now here we will deal with one special part of a circle that is sector. We shall discuss what is a sector? Problems on finding the perimeter of a circle sector and different formulas associated with it.

Sector 

A part of a circle that is formed by an arc and two radii of a circle is said to be the sector of a circle. From the below figure the colored area is called a sector. The sector can be assumed as a slice of a pizza.(images will be updated soon).

A sector is represented by sector OAB where angle ABC is θ and r represents the radius of the circle.

A circle is divided into two sectors, namely the major sector, and the minor sector.

Major Sector

A larger part of the circle occupied by the radii and the major arc is said to be a major sector.

In the figure the yellow colored part is the major sector.(image will be updated soon)

Minor Sector

A smaller part occupied by two radii and a minor arc is called the minor sector. 

In the figure, the blue-colored part is a minor sector.(image will be updated soon)

Area of Sector (image will be updated soon)

Consider a circle with O as center and r as the radii of the circle. Let OAB is the minor sector such that angle AOB = θ . θ is less than 1800.

Now if increases, the length of the arc AB also increases if = 1800 then arc AB becomes the circumference of a semi-circle that is πr.

Therefore,  Arc length l = θ/180 x πr

If the arc is subtended for the whole circle the length of the arc

Can also be written as, l = θ/360 x circumference of a circle

If the arc subtends an angle , then the area of the corresponding sector is 

Can also be written as, A =θ/360 x area of the circle

Perimeter of a Circle Sector

The perimeter of a circle sector is the sum of the two radii and the subtended arc length of a circle. Perimeter of a circle sector is also called the circumference of a sector.(image will be updated soon)

Consider the above figure, here the yellow colored part is the minor sector.

Perimeter of a Sector Formula

The perimeter of sector of circle formula is given below :

Perimeter of sector/circumference of a sector = 2 x radius + arc length

Arc length is also calculated as

Arc length = θ/360 x 2πr

Arc length = r x (2θπ/360 )

Arc length = radius × central angle

Therefore, the perimeter of a sector formula is also given as:

Let us understand how to calculate the perimeter of a sector or circumference of a sector by the solved example.

Solved Examples

Example 1: Find the area of a sector of a circle whose radius is 28cm and the central angle is 450.

Solution: We know that the area A of a sector of a circle is given by 

A = θ/360 x πr2

Here, r = 28 cm and θ= 450

A = 45/360 x 22/7 x 28 x 28

A = 308 cm2

Example 2: The perimeter of a sector of a circle of radius is 5.2cm is 16.4 cm. Find the area of the sector.

Solution: Perimeter of sector of circle formula is given by

Perimeter of sector = 2 radius + arc length

Here, r = 5.2 cm and perimeter of sector = 16.4 cm

16.4 = 2 x 5.2 + arc length

Arc length = 16.4 - 10.4

Arc length = 6cm

Also we have,

Area of sector = ½ lr

= ½ x 6 x 5.2

= 15.6cm2

Quiz Time