Table of Content

• What is Sector of a Circle?
• What is Segment of a Circle?
• Important Formulas-Area of Sector and Segment of a Circle
• Areas of Sector and Segment of a Circle Practice Problems
• Practice Questions on Areas of Sector and Segment of a Circle
• FAQs on Areas of Sector and Segment of a Circle

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°) × 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, θ_rad = (π × 73.74)/180 ≈ 1.29 radians.

Calculate area of sector

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

Calculate area of triangle formed by the chord using A_triangle = 1/2c (4r² – c²)^1/2

A_triangle = 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:

A_segment = A_sector – A_triangle = 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,

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• Area – Questions and Answers
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Frequently Asked Questions

What is Sector of a Circle?

A sector of a circle is a region enclosed by two radii of the circle and and the corresponding arc between them.

What is Segment of a Circle?

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

What are Real-World Applications of Areas of Sectors and Segments of Circles?

• Sectors: Used in engineering for calculating the area of sectors in circular motion applications, such as in parts of machinery.

• Segments: Used in construction for calculating the area of segments in circular concrete slabs or arches.

How do you Find Area of a Circular Sector given Central Angle?

To find the area of a sector given the central angle θ (in radians) use the formula:

A = 1/2 r² θ