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.
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.
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 |
|
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°) × r2θ we get:
A = (π/360°) × (8)2 × 45
A = (64 × 45 × π)/360
A = 8Ï€
Therefore, the area of the sector is 8Ï€ square centimeters.
Problem 2: A sector of a circle has a radius of 12 cm and an area of 36Ï€ square cm. Find the measure of the central angle of the sector.
Solution:
Given: Radius, r = 12 cm, Area of sector, A = 36Ï€ 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 = 100Ï€/3 square cm
Therefore, the area of the sector is 100Ï€/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 75Ï€ 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 36Ï€ 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 18Ï€ 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,