Areas of Sector and Segment of a Circle Practice Problems (original) (raw)

Last Updated : 23 Jul, 2025

Sector of a circle is a region bounded by two radii of the circle and the corresponding arc between these radii whereas a segment of a circle is the region bounded by a chord and the arc subtended by the chord.

In this article, we will learn about the area of sector and segment of a circle along with a few important formulas and various solved examples and practice questions on the area of sector and segment of a circle.

Table of Content

What is Sector of a Circle?

Sector of a circle is a fractional part of a circle, defined by a central angle and extending from the centre of the circle to its circumference.

The area of the sector of a circle is bounded by two circle radii and the arc on which these radii meet.

Definition of Sector

We can also say that the area of a sector is directly proportional to the square of the radius r and the central angle θ. **The larger the radius or the angle, the larger the sector's area.

What is Segment of a Circle?

Segment of a circle is the area bounded by the chord and the arc formed from the endpoint of the chord.

Definition of Segment

The **area of a segment is calculated by subtracting the area of the triangle formed by the chord from the area of the sector defined by the same chord.

Important Formulas-Area of Sector and Segment of a Circle

Few important formulas related to Area of Sector and Segment of a Circle are given below:

Term Formula
Area of Sector of a Circle A = (π/360°) × r2θ(when θ in degrees) A = (1/2) × r2θ (when θ in radians)
Arc Length of a Sector L = θ/360° × 2πr
Area of Segment (when θ in radians) (1/2) × r2(θ – sinθ)
Area of Segment (when θ in degrees) (1/2) × r2 [(π/180) θ – sinθ]
Area of Segment of a Circle Area of Sector – Area of Triangle

Areas of Sector and Segment of a Circle Practice Problems

**Problem 1: Calculate the area of a sector of a circle with radius 8 cm and a central angle of 45 ∘ .

**Solution:

Given, r = 8 cm and θ = 45∘

Putting the given values in the formula, **A = (π/360°) × r 2 θ we get:

A = (π/360°) × (8)2 × 45

A = (64 × 45 × π)/360

A = 8π

Therefore, the area of the sector is 8π square centimeters.

**Problem 2: A sector of a circle has a radius of 12 cm and an area of 36π square cm. Find the measure of the central angle of the sector.

**Solution:

Given: Radius, r = 12 cm, Area of sector, A = 36π square cm.

We know the formula to calculate the area of sector is given by: ( A = 1/2r2θ)

Putting the values in the above formula we get:

36π = 1/2 × (12)2 × θ

36π = 72 × θ

θ = 36π/72

θ = π/2 radians or 90°

**Problem 3: Calculate the area of a sector of a circle whose diameter is 20 cm, and the central angle is 120 ∘ .

**Solution:

Given diameter = 20cm and θ = 120∘ or (π × 120)/180 = 2π/3 radians

We know radius = diameter/2 = 20/2 = 10 cm

Also the formula to calculate the area of sector is given by: ( A = 1/2r2θ)

Putting the values in the above formula we get:

A = 1/2 × (10)2 × 2π/3

A = 1/2 × 100 × 2π/3

A = 100π/3 square cm

Therefore, the area of the sector is 100π/3 square cm.

**Problem 4: Calculate the area of a segment of a circle with radius 10 cm and chord length 12 cm.

**Solution:

Find the central angle θ that corresponds to the chord using, θ = 2sin-1 (c/2r)

Here, c = 12cm and r = 10cm, putting these values we get

θ = 2sin-1 (12/20) = 2 sin-1(0.6) ≈ 73.74∘

Now, Convert θ to radians, θrad = (π × 73.74)/180 ≈ 1.29 radians.

**Calculate area of sector

Asector = 1/2 × (10)2 × 1.29 = 64.5 square cm

**Calculate area of triangle formed by the chord using Atriangle = 1/2c (4r2 - c2)1/2

Atriangle = 1/2 × 12 × (4 × (10)2 - (12)2)1/2

= 6 × (256)1/2 = 96 square cm

**Calculate area of segment by subtracting the area of the triangle from the area of the sector:

Asegment = Asector - Atriangle = 64.5 - 96 = -31.5 square cm

**Problem 5: Find the area of the segment if the radius of the circle is 5 cm and subtended angle is π/6.

**Solution :

Area of segment (when θ in radians) = (1/2) × r2 [ θ – sinθ]

⇒ Area of segment = (1/2) × 52 [π/6 – sinπ/6]

⇒ Area of segment = (1/2) × 25[π/6 – 1/2]

⇒ Area of segment = (1/2) × 25[(3.14 - 3)/6]

⇒ Area of segment = (1/2) × 25 × (0.14)/6

⇒ Area of segment = 0.291 cm2

Practice Questions on Areas of Sector and Segment of a Circle

**Q1. Calculate the area of a sector of a circle with radius 12 cm and a central angle of 45∘.

**Q2. Find the area of a segment of a circle with radius 14 cm and a chord length of 16 cm.

**Q3. Calculate the area of a segment of a circle with radius 10 cm and a chord length of 12 cm.

**Q4. A sector of a circle has a radius of 15 cm. If the area of the sector is 75π square cm, find the central angle of the sector.

**Q5. A segment of a circle has a radius of 18 cm and a central angle of 120∘. Calculate the area of the segment.

**Q6. In a circle with radius 25 cm, the area of a segment is 150 square cm. Find the chord length of the segment.

**Q7. Find the area of a sector of a circle with radius 8 cm and a central angle of 120∘.

**Q8. The area of a sector of a circle is 36π square units. If the radius of the circle is 9 units, find the measure of the central angle in radians.

**Q9. In a circle with radius 6 cm, the area of a sector is 18π square cm. Find the central angle of the sector.

**Q10. The area of a segment of a circle is 64 square units. If the radius of the circle is 8 units and the central angle is 90∘, find the length of the chord.

**Read More,