Area of Segment of a Circle

A segment of a circle is the region enclosed between a chord and the arc that the chord cuts off. It is an important concept in geometry that combines the ideas of areas of sectors and triangles. In this guide, you will learn the definition of a segment of a circle, the formula to find the area of the segment, step-by-step derivation and solved problems for easy understanding.

Table of Contents

• Formula to Find Area of a Segment of a Circle
• Solved Examples on the Area of a Segment of a Circle
• Practice Questions on the Area of a Segment of a Circle

Formula to Find Area of a Segment Of a Circle

A segment of a circle is the region between a chord and the arc intercepted by that chord. The area of a segment is calculated using the area of the corresponding sector minus the area of the triangle formed by the chord and the two radii.

Area of Segment = Area of Sector − Area of Triangle
Area of Segment = (θ/360) × πr² − (1/2) × r² × sin θ

Where,

• θ = central angle of the sector (in degrees)

•  r = radius

If the θ is in radians:
Area of Sector = (1/2) r² θ.
Then, Area of Segment =   (1/2) r² θ - (1/2) × r² × sin θ
Area of Segment =  (1/2) r² (θ -  sin θ)

Solved Examples on the Area of a Segment of a Circle

Example 1: Find the area of the minor segment of a circle with radius 14 cm, where the central angle is 90°. (Use π = 22/7)
Solution: Given, r = 14 cm , θ =  90°
Area of the segment = (θ/360) × πr² − (1/2) × r² × sin θ
=(90°/360°) × 227 × 14² − (1/2) × 14² × sin 90°
= 154 - 98
= 56 cm².

Example 2: Find the area of the minor segment of a circle with radius 21 cm, where the central angle is 60°. (Use π = 22/7, √3 = 1.73)
Solution: Given, r = 21 cm and θ =  60°
Area of the segment = (θ/360) × πr² − (1/2) × r² × sin θ
=(60°/360°) × 227 × 21² − (1/2) × 21² × sin 60°
=  (1/6) × (22/7) × 441  -  (1/2) × (1.73/2) × 441
= 231 − 190.73
= 40.27 cm²

Example 3: A chord subtends an angle of 120° at the centre of a circle of radius 7 cm. Find the area of the corresponding minor segment. (Use π = 22/7, √3 = 1.73)
Solution: Given, r = 7 cm and θ =  120°
Area of the segment = (θ/360) × πr² − (1/2) × r² × sin θ
=(120°/360°) × 227 × 7² − (1/2) × 7² × sin 120°
=  (1/3) × (22/7) × 49  -  (1/2) × (1.73/2) × 49
= 51.33 − 21.19
= 30.14 cm²

Practice Questions on the Area of a Segment of a Circle

1. A circular tabletop of radius 35 cm has a segment cut off by a chord subtending 60° at the centre. Find the area of the remaining tabletop.

2. Find the area of the major segment of a circle with radius 7 cm and central angle 45°. (Use π = 22/7)

3. A chord subtends a right angle at the centre of a circle with radius 10 cm. Find the area of the minor segment. (Use π = 3.14)