Area of Sector PYQs

Class 10 Previous Year Questions (2014 – 2026)

Topic Overview

A sector is a portion of a circle enclosed by two radii and an arc. The area formula is $\frac{\theta}{360} \times \pi r^2$. Pay close attention to clock-hand problems, where the angle swept is $6^{\circ}$ per minute.

Q1 2026

00:00

The area of a sector of a circle with radius 14 cm and central angle 45° is:

(a)(A) 77 cm²

(b)(B) 154 cm²

(c)(C) 38.5 cm²

(d)(D) 308 cm²

Area $= (45/360) \times (22/7) \times 14 \times 14$.

Area $= 1/8 \times 22 \times 2 \times 14 = 77$ cm².

Ans: (A) 77 cm²

Q2 2025

00:00

If the area of a sector of a circle is 1/10 of the area of the circle, then the central angle is:

(a)(A) 18°

(b)(B) 36°

(c)(C) 72°

(d)(D) 90°

$(\theta/360) \times \pi r^2 = 1/10 \times \pi r^2$.

$\theta/360 = 1/10 \Rightarrow \theta = 36^{\circ}$.

Ans: (B) 36°

Q3 2024

00:00

The area of a sector of a circle with radius 6 cm if angle of the sector is 60° is:

(a)(A) $132/7$ cm²

(b)(B) $154/7$ cm²

(c)(C) $164/7$ cm²

(d)(D) $176/7$ cm²

Area of sector $= (\theta/360) \times \pi r^2$.

Area $= (60/360) \times (22/7) \times 6 \times 6 = (1/6) \times (22/7) \times 36 = 132/7$ cm².

Ans: (A) $132/7$ cm²

Q4 2023

00:00

The area of a sector of angle $p$ (in degrees) of a circle with radius R is:

(a)(A) $(p/180) \times 2\pi R$

(b)(B) $(p/180) \times \pi R^2$

(c)(C) $(p/360) \times 2\pi R$

(d)(D) $(p/720) \times 2\pi R^2$

Area $= (p/360) \times \pi R^2$.

Multiply numerator and denominator by 2: $(p/720) \times 2\pi R^2$.

Ans: (D)

Q5 2022

00:00

Find the length of the arc of a circle of radius 14 cm which subtends an angle of 90° at the center.

Length of arc $= (\theta/360) \times 2\pi r$.

Length $= (90/360) \times 2 \times (22/7) \times 14 = (1/4) \times 2 \times 22 \times 2 = 22$ cm.

Ans: 22 cm

Q6 2020

00:00

The angle described by a minute hand in 5 minutes is:

(a)(A) 30°

(b)(B) 60°

(c)(C) 90°

(d)(D) 15°

In 60 minutes, minute hand covers 360°.

In 1 minute, it covers $360/60 = 6^{\circ}$.

In 5 minutes, it covers $5 \times 6 = 30^{\circ}$.

Ans: (A) 30°

Q5 2024

00:00

Find the area of a quadrant of a circle whose circumference is 22 cm.

$2\pi r = 22 \Rightarrow 2 \times (22/7) \times r = 22 \Rightarrow r = 3.5$ cm.

Area of quadrant $= 1/4 \times \pi r^2 = 1/4 \times (22/7) \times 3.5 \times 3.5 = 9.625$ cm².

Ans: 9.625 cm²

Q6 2023

00:00

An arc of a circle is of length $5\pi$ cm and the sector it bounds has an area of $20\pi$ cm². Find the radius of the circle.

Area $= 1/2 \times l \times r$.

$20\pi = 1/2 \times 5\pi \times r$.

$40 = 5r \Rightarrow r = 8$ cm.

Ans: 8 cm

Q7 2022

00:00

The minute hand of a clock is 14 cm long. Find the area swept by the minute hand in 5 minutes.

Radius $r = 14$ cm. $\theta$ in 5 mins $= 30^{\circ}$.

Area $= (30/360) \times (22/7) \times 14 \times 14$.

Area $= (1/12) \times 22 \times 2 \times 14 = (1/6) \times 22 \times 14 = 154/3 = 51.33$ cm².

Ans: 51.33 cm²

Q8 2019

00:00

A sector of a circle of radius 12 cm has an angle of 120°. Find the length of its arc and its area. (Use $\pi = 3.14$)

Length $= (120/360) \times 2 \times 3.14 \times 12 = 1/3 \times 75.36 = 25.12$ cm.

Area $= (120/360) \times 3.14 \times 12 \times 12 = 1/3 \times 452.16 = 150.72$ cm².

Ans: Length = 25.12 cm, Area = 150.72 cm²

Q9 2025 (SP)

00:00

PASSAGE: Car Wipers. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°.
(i) Find the area cleaned by one wiper.
(ii) Find the total area cleaned by both wipers at each sweep.
(iii) (a) Find the length of arc covered by one blade. OR (b) If angle increases to 120°, find the increase in area.

(i) Area $= (115/360) \times \pi \times 25^2 \approx 627.48$ cm².

(ii) Total Area $= 2 \times 627.48 = 1254.96$ cm².

(iii) (a) Arc $= (115/360) \times 2 \times \pi \times 25 \approx 50.17$ cm.

Q10 2024

00:00

PASSAGE: Umbrella Design. An umbrella has 8 ribs which are equally spaced. Assuming umbrella to be a flat circle of radius 45 cm.
(i) Find the angle between two consecutive ribs.
(ii) Find the area between two consecutive ribs.
(iii) Find the total area of the flat circle.

(i) $\theta = 360/8 = 45^{\circ}$.

(ii) Area $= (45/360) \times (22/7) \times 45^2 = 1/8 \times 6364.28 = 795.53$ cm².

(iii) Total Area $= \pi \times 45^2 = 6364.28$ cm².

00:00

The area of a sector of a circle with radius 14 cm and central angle 45° is:

(a)(A) 77 cm²

(b)(B) 154 cm²

(c)(C) 38.5 cm²

(d)(D) 308 cm²

Area $= (45/360) \times (22/7) \times 14 \times 14$.

Area $= 1/8 \times 22 \times 2 \times 14 = 77$ cm².

Ans: (A) 77 cm²

Q2 2025

00:00

If the area of a sector of a circle is 1/10 of the area of the circle, then the central angle is:

(a)(A) 18°

(b)(B) 36°

(c)(C) 72°

(d)(D) 90°

$(\theta/360) \times \pi r^2 = 1/10 \times \pi r^2$.

$\theta/360 = 1/10 \Rightarrow \theta = 36^{\circ}$.

Ans: (B) 36°

Q3 2024

00:00

The area of a sector of a circle with radius 6 cm if angle of the sector is 60° is:

(a)(A) $132/7$ cm²

(b)(B) $154/7$ cm²

(c)(C) $164/7$ cm²

(d)(D) $176/7$ cm²

Area of sector $= (\theta/360) \times \pi r^2$.

Area $= (60/360) \times (22/7) \times 6 \times 6 = (1/6) \times (22/7) \times 36 = 132/7$ cm².

Ans: (A) $132/7$ cm²

Q4 2023

00:00

The area of a sector of angle $p$ (in degrees) of a circle with radius R is:

(a)(A) $(p/180) \times 2\pi R$

(b)(B) $(p/180) \times \pi R^2$

(c)(C) $(p/360) \times 2\pi R$

(d)(D) $(p/720) \times 2\pi R^2$

Area $= (p/360) \times \pi R^2$.

Multiply numerator and denominator by 2: $(p/720) \times 2\pi R^2$.

Ans: (D)

Q5 2022

00:00

Find the length of the arc of a circle of radius 14 cm which subtends an angle of 90° at the center.

Length of arc $= (\theta/360) \times 2\pi r$.

Length $= (90/360) \times 2 \times (22/7) \times 14 = (1/4) \times 2 \times 22 \times 2 = 22$ cm.

Ans: 22 cm

Q6 2020

00:00

The angle described by a minute hand in 5 minutes is:

(a)(A) 30°

(b)(B) 60°

(c)(C) 90°

(d)(D) 15°

In 60 minutes, minute hand covers 360°.

In 1 minute, it covers $360/60 = 6^{\circ}$.

In 5 minutes, it covers $5 \times 6 = 30^{\circ}$.

Ans: (A) 30°

Q5 2024

00:00

Find the area of a quadrant of a circle whose circumference is 22 cm.

$2\pi r = 22 \Rightarrow 2 \times (22/7) \times r = 22 \Rightarrow r = 3.5$ cm.

Area of quadrant $= 1/4 \times \pi r^2 = 1/4 \times (22/7) \times 3.5 \times 3.5 = 9.625$ cm².

Ans: 9.625 cm²

Q6 2023

00:00

An arc of a circle is of length $5\pi$ cm and the sector it bounds has an area of $20\pi$ cm². Find the radius of the circle.

Area $= 1/2 \times l \times r$.

$20\pi = 1/2 \times 5\pi \times r$.

$40 = 5r \Rightarrow r = 8$ cm.

Ans: 8 cm

Q7 2022

00:00

The minute hand of a clock is 14 cm long. Find the area swept by the minute hand in 5 minutes.

Radius $r = 14$ cm. $\theta$ in 5 mins $= 30^{\circ}$.

Area $= (30/360) \times (22/7) \times 14 \times 14$.

Area $= (1/12) \times 22 \times 2 \times 14 = (1/6) \times 22 \times 14 = 154/3 = 51.33$ cm².

Ans: 51.33 cm²

Q8 2019

00:00

A sector of a circle of radius 12 cm has an angle of 120°. Find the length of its arc and its area. (Use $\pi = 3.14$)

Length $= (120/360) \times 2 \times 3.14 \times 12 = 1/3 \times 75.36 = 25.12$ cm.

Area $= (120/360) \times 3.14 \times 12 \times 12 = 1/3 \times 452.16 = 150.72$ cm².

Ans: Length = 25.12 cm, Area = 150.72 cm²

Q9 2025 (SP)

00:00

(i) Area $= (115/360) \times \pi \times 25^2 \approx 627.48$ cm².

(ii) Total Area $= 2 \times 627.48 = 1254.96$ cm².

(iii) (a) Arc $= (115/360) \times 2 \times \pi \times 25 \approx 50.17$ cm.

Q10 2024

00:00

(i) $\theta = 360/8 = 45^{\circ}$.

(ii) Area $= (45/360) \times (22/7) \times 45^2 = 1/8 \times 6364.28 = 795.53$ cm².

(iii) Total Area $= \pi \times 45^2 = 6364.28$ cm².