Area Related to Circles Class 10 Notes Maths Chapter 11

Introduction

Sector of Circle: The area of a circular region that is bounded by two radii and the arc between them is known as a sector of the circle.

• The portion OAPB of the circle is called the minor sector and the portion OAQB of the circle is called the major sector. 
• ∠ AOB is called the angle of the sector.

Arc: An arc is a portion of the circle's circumference.

Chord: A chord is a line segment that joins any two points on the circle's circumference.

Segment of Circle: The area of a circular region that lies between a chord and the corresponding arc is referred to as a segment of the circle.

• A minor segment is made by a minor arc.
•  A Major segment is made by a major arc of the circle.

Area of a Sector of a Circle

Sector: Sector of a Circle is given as part of a Circle enclosed by 2 radii and an arc.

In the diagram, the shaded area OAB is the sector.
Here, θ is the angle subtended by the arc AB on the center O of the circle.

The area of the Sector is given as  

In the whole circle, the angle θ will be 360°
Area of Circle = πr²
Using Unitary Method
Area represented by 360° = πr²
Area represented by

Length of an Arc of a sector of angle θ  =

Solved Examples

Q1: Calculate the area of a sector with a radius of 20 yards and an angle of 90 degrees.

Ans: here θ = 90º, r = 20 yards, π = 3.141

= (90º/360º) X 3.141 X (20)²

= (1256.4/4) yards² 

= 314.1 yards² 

Q2: The area of a sector is 225 m². If the sector’s radius is 8 m, find the central angle of the sector in radians.

Ans: 

Area of Segment of a circle 

In the diagram Shaded portion represents Segment AMB

Area of Segment AMB = Area of Sector OAB- Area of triangle AOB

Solved Examples 

Q1: Given that a chord and radius of a circle are each 24 cm. Find the area of the minor circular segment.

Ans: From the diagram, it is clear that ΔOBC is an equilateral triangle.

Hence, the central θ is 60º = π/3 radians

As we know, 

Area (A) of a Segment of a Circle in radians = 1/2 X r² (θ - Sinθ), here

r = 22cm, θ = v

= 1/2 X (24)² [(π/3 - Sin(π/3)]

= 52.18 cm²

Q2: Find the area of the major segment of a circle if the area of the corresponding minor segment is 88 m² and the radius is 22 m. Use π = 3.141.

Ans: Area of the major segment = Area of the circle - Area of the minor segment 

= πr² - 88

= [3.141 x (22)²] - 88

= 1432.24 m²