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


l = θ/360 x 2πr


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 


A = θ/360 x πr2


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


Perimeter of sector = 2 r  + 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:


Perimeter of a sector = 2Radius + (radius x central angle)


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

    1. In a circle with o as center and radius is 21cm. Area of sector is 577.5cm2.Find the length of the arc.
    2. A sector is cut from the circle with radius 21cm. The angle of the sector is 1500. Find the length of the arc and its area.
    FAQ (Frequently Asked Questions)

    1.What is the Circumference of the Circle and how It is Calculated?

    Answer : The distance around the circle or you can say the boundary of the circle is said to be circumference of a circle. Circumference can also be said as the length of a circle.(image will be updated soon)


    The circumference of a circle is calculated by just multiplying pi with the diameter of the circle. Pi is represented as and it has the constant value of 22/7 or  3.141592653589793  or simply 3.14.

     π is defined as the ratio of circumference of a circle with the diameter. For any circle the value of π is almost the same.

    Therefore to find the circumference of a circle we have the formula

    Circumference of a circle = πd, where d is the diameter of the circle.

    It can be written in the form of radius as 

    Circumference of a circle = π 2r = 2πr.

    2. What is the Segment of a Circle?

    Answer: When a chord of a circle divides the circle into two regions, then the region enclosed by an arc and a chord is said to be a segment of the circle. A chord divides the circle into two segments, major segment, and minor segment.

    If the boundary of a segment is a minor arc of a circle, then the corresponding segment is called a minor segment.

    A segment corresponding to the major arc of a circle is known as a major segment.

    In the figure below ACB is a minor segment and ADB is a major segment.(image will be updated soon).