Areas of Sector and Segment of a Circle Practice Problems
Last Updated :
14 Jul, 2024
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.
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.
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 |
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Area of Segment (when θ in radians) | (1/2) × r2(θ – sinθ) |
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Area of Segment (when θ in degrees) | (1/2) × r2 [(π/180) θ – sinθ] |
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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 = 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.
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