Areas Related to Circles Class 10 Previous Year Questions: 1 Mark (Chapter 11)

1-mark questions on Areas Related to Circles

This page has 25 1-mark questions on Areas Related to Circles from CBSE Class 10 board papers of 2023, 2024, 2025 and 2026. Of these, 13 appeared only in Standard papers and 12 appeared only in Basic papers. Question formats: 25 MCQs. 11 of them were asked in two or more paper sets.

0 of 25 practised

All 25 Areas Related to Circles 1-mark questions

Q1. The angle of the sector of a circle whose area is one-eighth of the area of the circle is :

Mark practised

(A) \(22\frac{1}{2}^\circ\)
(B) \(45^\circ\)
(C) \(60^\circ\)
(D) \(90^\circ\)

Marks: 1
Level: Basic
Asked in: 6 paper sets
Years: 2025, 2026
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2025 Mathematics Basic, Set 430/3/1, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 45°, because area of a sector = (θ/360) × πr², so one-eighth of 360° is 360/8 = 45.
Why? The area of a sector varies directly with its angle at the centre, so equal fractions of the circle's area correspond to the same fractions of 360°.

The full 1 mark is for the correct option.

Answer:

(B) 45°

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/3/1: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 22½° by taking one-eighth of 180° instead of 360°.
Choosing 90° because a quarter-circle has angle 90°, but the sector is one-eighth, not one-quarter.
Choosing 60° by splitting 360° into 6 equal parts, but the sector is 1/8 of the circle, not 1/6.

Why this works

The whole circle has angle 360° and area πr². A sector of angle θ has area (θ/360) × πr². The question says this sector area is one-eighth of the circle's area, so (θ/360) × πr² = (1/8) × πr². Cancelling πr² gives θ/360 = 1/8. Multiplying both sides by 360 gives θ = 360/8 = 45. Hence the angle is 45°, option (B). Another correct method: because area is proportional to the angle, one-eighth of the circle's area means one-eighth of 360°, that is 360/8 = 45.

Key idea to remember

Area of a sector of a circle: see Key ideas used on this page.

Explanation

The formula Area of the sector of angle θ = (θ/360) × πr² shows that a sector's area is the fraction θ/360 of the whole circle's area πr². This solution sets that fraction equal to 1/8 and solves for θ = 360/8 = 45.

Important formulas

Area of the sector of angle θ = (θ/360) × πr² · θ is the angle of the sector in degrees and r is the radius of the circle.

Q2. OACB is a quadrant of a circle with centre O and radius 7 cm where ACB is the arc. Then the perimeter of the quadrant is :

Mark practised

(A) 15 cm
(B) 50 cm
(C) 25 cm
(D) 44 cm

Marks: 1
Level: Standard
Asked in: 3 paper sets
Compartment: 2023
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2023 Compartment Mathematics Standard, Set 30/C/1, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 25 cm, because the perimeter of the quadrant = arc ACB + OA + OB = (90/360) × 2 × (22/7) × 7 + 7 + 7 = 11 + 14 = 25 cm.
Why? A quadrant is a sector of angle 90°, so its arc is one-fourth of the circumference, and the perimeter of the quadrant is the arc plus the two radii.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Compartment, Paper 30/C/1: option (C).

Open CBSE’s official marking scheme for the 2023 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Forgetting to add the two radii OA and OB, and giving only the arc length 11 cm as the perimeter.
Choosing 44 cm, the whole circumference of the circle, instead of the quadrant's arc plus the two radii.
Using 3.14 for π, getting 24.99 cm, and not recognising that the option 25 cm is the same value.

Why this works

Perimeter means the length of the whole boundary. For the quadrant OACB, the boundary is the curved arc ACB together with the two straight radii OA and OB. A quadrant makes an angle of 90° at the centre, which is one-fourth of 360°, so its arc is one-fourth of the circumference. Using the value 22/7 for π, the whole circumference is 2 × (22/7) × 7 = 44 cm, the arc is 44 ÷ 4 = 11 cm, and the perimeter is 11 + 7 + 7 = 25 cm. So option (C) is correct. Option (D) 44 cm is the whole circumference. Option (A) 15 cm is too small because the two radii alone are 14 cm and the arc is 11 cm. Option (B) 50 cm does not match arc + two radii.

Key idea to remember

Length of an arc of a sector of a circle: see Key ideas used on this page.

Explanation

A quadrant is a sector whose angle at the centre is 90°. The NCERT formula gives the length of an arc of a sector of angle θ as (θ/360) × 2πr. With the angle 90°, the arc is one-fourth of the circumference. The perimeter of the quadrant is this arc plus the two radii, because the boundary of the region OACB is OA, arc ACB and OB.

Important formulas

Length of an arc of a sector of angle θ = θ/360 × 2πr · For a quadrant, the angle is 90°, so the arc is one-fourth of the circumference.
Perimeter of a quadrant = arc length + 2r · The boundary of the quadrant has two radii and one arc.

Q3. What is the length of the arc of the sector of a circle with radius 14 cm and of central angle 90° ?

Mark practised

(A) 22 cm
(B) 44 cm
(C) 88 cm
(D) 11 cm

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2023
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2023 Mathematics Standard, Set 30/2/1, Q2, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 22 cm, because (90/360) × 2 × (22/7) × 14 = 22 cm.
Why? The whole circle has arc length 2πr for an angle of 360°, so the arc for angle θ is the fraction θ/360 of the circumference.

The full 1 mark is for the correct option.

Answer:

(A) 22 cm

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/2/1: option (A).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the area formula (90/360) × (22/7) × 14², which gives 154 cm², instead of the arc length formula.
Writing the whole circumference 88 cm as the answer, because the 90° angle was not converted to the fraction 1/4.
Taking π = 3.14, which gives 21.98 cm and does not match an option, instead of π = 22/7 used in this exercise.

Why this works

A central angle of 90° is one quarter of a full turn of 360°. So the arc is one quarter of the whole circumference. With π = 22/7, the circumference is 2 × (22/7) × 14 = 88 cm, and one quarter of it is 88/4 = 22 cm. This is exactly what the formula (90/360) × 2 × (22/7) × 14 = 22 cm gives.

Key idea to remember

Length of an arc of a sector of a circle: see Key ideas used on this page.

Explanation

The NCERT formula says: length of an arc of a sector of a circle with radius r and angle with degree measure θ is θ/360 × 2πr. The fraction θ/360 tells what part of the whole circumference the arc is, because the whole circle corresponds to an angle of 360°.

Important formulas

l = θ/360 × 2πr · Length of an arc of a sector of a circle with radius r and angle θ in degrees.

Q4. If length of an arc of a circle subtending an angle \(\theta\) at the centre is numerically equal to the area of the sector formed by it, then the radius of the circle is :

Mark practised

(A) 1 unit
(B) 2 units
(C) 3 units
(D) \(\frac{1}{2}\) unit

Marks: 1
Level: Basic
Asked in: 3 paper sets
Compartment: 2024
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2024 Compartment Mathematics Basic, Set 430/S/1, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 2 units, because θ/360 × 2 × π × r = θ/360 × π × r², and cancelling θ/360 × π × r gives r = 2.
Why? Equating the two NCERT formulae and cancelling the common positive factor θ/360 × πr leaves the radius equal to 2.

The full 1 mark is for the correct option.

Answer:

(B) 2 units

✓ Answer matches the CBSE answer key, 2024 Compartment, Paper 430/S/1: option (B).

Open CBSE’s official marking scheme for the 2024 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 1 unit because 2πr looks like twice πr², without cancelling the common factor πr first.
Dividing both sides by r without noting r ≠ 0; a circle's radius is positive, so r = 0 is not a valid solution.
Thinking θ must be a particular value such as 180° to make the two expressions equal, instead of letting θ cancel.

Why this works

The question gives no value of θ, so θ must cancel out. Write the two formulae: length of an arc is θ/360 × 2πr and area of a sector is θ/360 × πr². Since the length is numerically equal to the area, θ/360 × 2πr = θ/360 × πr². The factor θ/360 × πr is common on both sides and is not zero because θ > 0 and r > 0, so cancel it: the radius is 2. Quick check with θ as 60°: arc length is 60/360 × 2π × 2, which is 2π/3, and sector area is 60/360 × π × 4, which is also 2π/3, so they are equal. The other options give unequal values: for r = 1, arc length is θ/360 × 2π but area is θ/360 × π; for r = 3, arc length is θ/360 × 6π but area is θ/360 × 9π; for r = 1/2, arc length is θ/360 × π but area is θ/360 × π/4.

Key idea to remember

Length of an arc of a sector of a circle and area of a sector of a circle: see Key ideas used on this page.

Explanation

The NCERT gives the length of an arc of a sector of angle θ as θ/360 × 2πr and the area of the sector as θ/360 × πr². This question sets the two expressions numerically equal and solves for r.

Important formulas

Length of an arc of a sector of angle θ = θ/360 × 2πr · where r is the radius and θ is the angle in degrees.
Area of a sector of angle θ = θ/360 × πr² · where r is the radius and θ is the angle in degrees.

Q5. The region between a chord and either of the two arcs of a circle is called :

Mark practised

(A) an arc
(B) a sector
(C) a segment
(D) a semicircle

Marks: 1
Level: Basic
Asked in: 3 paper sets
Years: 2024
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2024 Mathematics Basic, Set 430/4/1, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) a segment, because the region between a chord and either of its arcs is exactly the segment of a circle.
Why? The term 'segment' is fixed by its boundary: one straight chord and one curved arc, unlike a sector whose boundary is two radii and an arc.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/4/1: option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 'a sector' because both are parts of a circular region, but a sector is enclosed by two radii and the corresponding arc, not by a chord.
Choosing 'an arc' because the curved boundary of the region is an arc, but the arc itself is only the line, not the enclosed region.
Choosing 'a semicircle' because a chord with an arc can look like half a circle, but a semicircle requires the chord to be a diameter.

Why this works

A chord cuts the circular region into two parts, one on each side of the chord. Each part is bounded by the chord and one of the two arcs, so each part is a segment. An arc is only the curved boundary, not the region itself. A sector is bounded by two radii and the corresponding arc, so it does not use a chord. A semicircle is the special case where the chord is a diameter. Therefore the region described is called a segment.

Key idea to remember

Segment of a circle: see Key ideas used on this page.

Explanation

The portion of the circular region enclosed between a chord and the corresponding arc is called a segment of the circle. For a given chord, the two arcs give two segments: the minor segment and the major segment. Unless stated otherwise, the word 'segment' means the minor segment.

Q6. The perimeter of the sector of a circle of radius 21 cm which subtends an angle of \(60^\circ\) at the centre of circle, is :

Mark practised

(A) 22 cm
(B) 43 cm
(C) 64 cm
(D) 462 cm

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2024
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2024 Mathematics Standard, Set 30/5/1, Q8, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 64 cm, because the perimeter of the sector = arc length + 2r = (60/360 × 2 × 22/7 × 21) + 2 × 21 = 22 + 42 = 64 cm.
Why? By the NCERT formula, the length of an arc of a sector of angle θ is θ/360 × 2πr; the perimeter is this arc plus the two radii.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/5/1: option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 22 cm, which is only the arc length, without adding the two radii.
Choosing 43 cm, which adds only one radius to the arc length instead of both radii.
Choosing 462 cm, which is the area of a 120° sector of radius 21 cm in NCERT Example 2, not the perimeter of a 60° sector.

Why this works

Perimeter means the total length of the boundary. A sector is bounded by two radii and the corresponding arc, so its perimeter is the arc length plus two radii. The NCERT chapter says to take π as 22/7 unless stated otherwise. Here the arc length = 60/360 × 2 × 22/7 × 21 = 22 cm, and the two radii add 2 × 21 = 42 cm. So the perimeter = 22 + 42 = 64 cm, which is option C. Quick check: circumference = 2 × 22/7 × 21 = 132 cm; one-sixth of it is 22 cm; adding 42 cm gives 64 cm. Option A is only the arc length, option B adds only one radius, and option D is the area of the 120° sector of radius 21 cm from NCERT Example 2, not this perimeter.

Key idea to remember

Length of an arc of a sector of a circle: see Key ideas used on this page.

Explanation

NCERT gives the length of an arc of a sector of angle θ as θ/360 × 2πr. A sector is the region enclosed by two radii and the corresponding arc, so its perimeter is the sum of this arc and the two radii.

Important formulas

Length of an arc of a sector = θ/360 × 2πr · θ is the angle of the sector in degrees, r is the radius.
Perimeter of a sector = length of the arc + 2r · The boundary of a sector has one arc and two radii.

Q7. A piece of wire 20 cm long is bent into the form of an arc of a circle of radius 60/π cm. The angle subtended by the arc at the centre of the circle is :

Mark practised

(A) 30°
(B) 60°
(C) 90°
(D) 50°

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2025
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2025 Mathematics Standard, Set 30/1/3, Q8, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 60°, because (θ/360) × 2π × (60/π) = 20 gives θ = 60°.
Why? By the formula for the length of an arc of a sector, l = (θ/360) × 2πr; the wire's length is the arc length, so substituting the given radius makes θ the only unknown.

The full 1 mark is for the correct option.

Answer:

(B) 60°

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/1/3: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the sector area formula (θ/360) × πr² instead of the arc-length formula; the area formula uses r², while the arc length uses r, so it cannot give the correct angle.
Forgetting the factor 2 in 2πr, so the full-circle length is taken as πr = 60 instead of 2πr = 120; this leads to 120° instead of 60°.
Taking the wire's length as the circumference of the circle; then the angle is not determined by the given radius.

Why this works

The wire is bent into the arc, so its whole length is the arc length. Put r = 60/π and l = 20 in the arc-length formula l = (θ/360) × 2πr. First, 2πr = 2π × (60/π) = 120. So (θ/360) × 120 = 20, which gives θ = 360 × 20 ÷ 120 = 60°. Check: with θ = 60°, the arc length is (60/360) × 2π × (60/π) = (1/6) × 120 = 20 cm, the same as the wire.

Key idea to remember

Length of an arc of a sector of a circle: see Key ideas used on this page.

Explanation

The NCERT text gives the length of an arc of a sector of a circle with radius r and angle with degree measure θ as (θ/360) × 2πr. Here the wire becomes the arc, so the arc length is 20 cm and the radius is 60/π cm. Substituting these values and solving gives θ = 60°.

Important formulas

l = (θ/360) × 2πr · Length of an arc of a sector of angle θ (in degrees) of a circle of radius r.

Q8. If a large circular pizza is divided into 5 equal sectors, then the central angle of each sector will be :

Mark practised

(A) \(60^\circ\)
(B) \(90^\circ\)
(C) \(45^\circ\)
(D) \(72^\circ\)

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2025
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2025 Mathematics Standard, Set 30/2/1, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 72°, because the whole circle makes 360° at the centre and 360° ÷ 5 = 72°.
Why? Equal sectors divide the total angle at the centre of a circle, which is 360°, equally among the number of sectors.

The full 1 mark is for the correct option.

Answer:

(D) 72°

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/2/1: option (D).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 60° by splitting only 300° of the circle, instead of the full 360°.
Dividing 180° by 5 and choosing 36°, treating the angle at the centre as a semicircle.
Choosing 90° because 90° × 4 = 360°, which is the angle for 4 sectors, not 5.

Why this works

A full circle makes an angle of 360° at the centre. When the pizza is cut into 5 equal sectors, each sector gets the same part of that 360°. So the central angle of one sector = 360° ÷ 5 = 72°. Quick check: 72° × 5 = 360°, so the five sectors exactly fill the circle. The other options would need a different number of sectors: 60° × 6 = 360° means 6 sectors, 90° × 4 = 360° means 4 sectors, and 45° × 8 = 360° means 8 sectors.

Key idea to remember

Angle of a sector: see Key ideas used on this page.

Explanation

A sector is the part of a circular region enclosed by two radii and the corresponding arc. The angle between the two radii at the centre is called the angle of the sector. The whole circle has angle 360°, and equal sectors divide this total equally, so each equal sector has angle 360° ÷ number of sectors.

Important formulas

Angle of one equal sector = 360° ÷ n · Here n is the number of equal sectors into which the circle is divided.

Q9. The length of arc subtending an angle of \(210^\circ\) at the centre of the circle, is \(\frac{44}{3}\) cm. The radius of the circle is :

Mark practised

(A) \(2\sqrt{2}\) cm
(B) 4 cm
(C) 8 cm
(D) \(\frac{1}{4}\) cm

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2025
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2025 Mathematics Basic, Set 430/4/1, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 4 cm, because (210/360) × 2 × (22/7) × r = 44/3, and solving gives r = 4.
Why? By the formula for the length of an arc of a sector, the arc length equals the angle fraction times the circumference; substituting the given values gives a linear equation in r.

The full 1 mark is for the correct option.

Answer:

(B) 4 cm

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/4/1: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the area of a sector formula (θ/360)πr² instead of the arc length formula; the area has units cm², but the question asks for a length in cm.
Using π = 3.14 instead of π = 22/7; the chapter directs the use of 22/7 unless stated otherwise, and the given arc length 44/3 is chosen to match 22/7.
After getting (11/3)r = 44/3, multiplying by the coefficient 11/3 instead of dividing by it, which gives a radius much larger than 4 cm.

Why this works

The whole circumference 2πr belongs to a full angle of 360°. So an arc whose angle at the centre is θ is the fraction θ/360 of the whole circumference. Here θ = 210°, so the arc is 210/360 = 7/12 of the circumference. With π = 22/7, 7/12 × 2 × 22/7 = 11/3, so the arc length is (11/3)r. The question gives this length as 44/3 cm, so (11/3)r = 44/3, giving r = 4 cm. Check: (11/3) × 4 = 44/3 cm.

Key idea to remember

Length of an arc of a sector of a circle: see Key ideas used on this page.

Explanation

The NCERT derives this by the unitary method: a full circle of angle 360° has circumference 2πr, so an angle of 1° gives an arc of length 2πr/360, and an angle of θ gives θ × 2πr/360. This solution substitutes the given angle 210° and the given arc length 44/3 cm to find r.

Important formulas

Length of an arc of a sector of angle θ = θ/360 × 2πr · θ is the angle at the centre in degrees; r is the radius of the circle.

Q10. The length of the arc of the sector of a circle with radius 21 cm and of central angle \(60^\circ\) , is :

Mark practised

(A) 22 cm
(B) 44 cm
(C) 88 cm
(D) 11 cm

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2026
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2026 Mathematics Standard, Set 30/1/1, Q14, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 22 cm, because length of the arc = 60/360 × 2 × 22/7 × 21 = 22 cm.
Why? The arc of a sector is the fraction θ/360 of the whole circumference 2πr.

The full 1 mark is for the correct option.

Answer:

(A) 22 cm

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/1/1: option (A).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing option D, 11 cm, by using πr (half the circumference) in place of 2πr.
Choosing option B, 44 cm, by putting the diameter 42 cm in place of the radius 21 cm.
Using the area formula θ/360 × πr² when the question asks for the length of the arc.

Q11. The hour hand of a clock is 7 cm long. The angle swept by it between 7:00 a.m. and 8:10 a.m. is :

Mark practised

(A) \(\left(\frac{35}{4}\right)^\circ\)
(B) \(\left(\frac{35}{2}\right)^\circ\)
(C) \(35^\circ\)
(D) \(70^\circ\)

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2026
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2026 Mathematics Standard, Set 30/1/1, Q15, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 35°, because the hour hand sweeps 0.5° per minute, and 7:00 a.m. to 8:10 a.m. is 70 minutes, so 0.5 × 70 = 35°.
Why? The hour hand completes 360° in 12 hours, so in 1 minute it turns 360/(12×60) = 0.5°; the length of the hand does not affect the angle.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/1/1: option (C).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using 7 cm to find an arc length or area, even though the question asks only for the angle swept.
Taking the interval as 1.10 hours and computing 30 × 1.10 as if it were 33°, instead of converting 1 hour 10 minutes to 70 minutes.
Working with the minute hand instead of the hour hand, so the speed or the time used is wrong.

Why this works

From 7:00 a.m. to 8:10 a.m. is 1 hour 10 minutes, which is 70 minutes. In 1 hour the hour hand turns 360°/12 = 30°, and in the next 10 minutes it turns 30° × 10/60 = 5°. So the total angle is 30° + 5° = 35°. Equivalently, 0.5° × 70 = 35°. The 7 cm length is extra information: the angle swept by a hand of a clock does not depend on the hand's length.

Key idea to remember

Unitary Method: see Key ideas used on this page.

Explanation

The chapter uses the unitary method for sectors: find the value for 1 degree, then multiply by the angle. This question does the same for time: find the angle turned in 1 minute, then multiply by the number of minutes.

Important formulas

Angle in 1 minute = 360° / (12 × 60) = 0.5° · The hour hand takes 12 hours to complete one full turn.
Angle swept = 0.5° × number of minutes · Multiply by the time in minutes between the two given times.

Q12. What is the length of arc of a circle of radius 7 cm which subtends an angle of \(90^\circ\) at the centre of the circle ?

Mark practised

(A) 22 cm
(B) 11 cm
(C) \(\frac{77}{2}\) cm
(D) \(\frac{11}{2}\) cm

Marks: 1
Level: Basic
Years: 2023
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2023 Mathematics Basic, Set 430/4/3, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 11 cm, because length of arc = (90/360) × 2 × (22/7) × 7 = 11 cm.
Why? A 90° angle is one-fourth of a full turn of 360°, so the arc is one-fourth of the whole circumference; this is the rule length of arc = (θ/360) × 2πr.

The full 1 mark is for the correct option.

Answer:

(B) 11 cm

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/4/3: option (B).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 22 cm because it is the arc for 180°, not for 90°.
Using the area of the sector formula (θ/360) × πr² and choosing 77/2 cm, which is an area, not a length.
Forgetting the factor 2 in 2πr and getting 11/2 cm.

Why this works

The whole circle has circumference 2πr = 2 × (22/7) × 7 = 44 cm. An angle of 90° at the centre is one-fourth of the full turn, so the arc is 1/4 of the circumference: 1/4 × 44 = 11 cm. This is exactly the formula length of arc = (θ/360) × 2πr. Since the question does not give a value of π, NCERT instructs us to use π = 22/7. Option A, 22 cm, is the arc for 180°; option C, 77/2 cm, is the area of the sector; option D, 11/2 cm, comes from forgetting the factor 2 in 2πr.

Key idea to remember

Length of an arc of a sector of a circle: see Key ideas used on this page.

Explanation

An arc is a part of the circumference of a circle. For a full circle the angle at the centre is 360° and the length is 2πr. By the unitary method, for an angle θ the arc length is θ/360 of the full circumference, which gives (θ/360) × 2πr. Here θ = 90° and r = 7 cm, so the arc is one-fourth of the circumference.

Important formulas

Length of an arc of a sector of angle θ = (θ/360) × 2πr · θ is in degrees and r is the radius of the circle.

Q13. If the perimeter and the area of a circle are numerically equal, then the radius of the circle is :

Mark practised

(A) 2 units
(B) \(\pi\) units
(C) 4 units
(D) \(2\pi\) units

Marks: 1
Level: Basic
Years: 2023
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2023 Mathematics Basic, Set 430/4/3, Q4, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 2 units, because 2πr = πr², so r = 2 (r > 0).
Why? For a circle, the perimeter is 2πr and the area is πr². Equating them and cancelling the positive factor πr gives the radius.

The full 1 mark is for the correct option.

Answer:

(A) 2 units

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/4/3: option (A).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 2π units because the perimeter formula contains 2π, without solving 2πr = πr².
Thinking the units must match; the perimeter is in units and the area is in square units, but the question compares only the numbers.

Why this works

Numerically equal means the two numbers are equal, even though the units are different. Let the radius be r units. Perimeter = 2πr and area = πr². Since r > 0, divide both sides of 2πr = πr² by πr to get 2 = r. Check: for r = 2, perimeter = 2π × 2 = 4π and area = π × 2² = 4π, so the numbers match. The other options fail: r = π gives 2π² ≠ π³, r = 4 gives 8π ≠ 16π, and r = 2π gives 4π² ≠ 4π³.

Key idea to remember

Circumference and area of a circle: see Key ideas used on this page.

Explanation

The perimeter of a circle, also called its circumference, is the distance around it and equals 2πr. The area of a circle is the space inside it and equals πr². When the question says they are numerically equal, it means 2πr = πr². Solving this equation gives the required radius.

Important formulas

Perimeter (circumference) of a circle = 2πr · r is the radius of the circle.
Area of a circle = πr² · r is the radius of the circle.

Q14. In a circle of radius 21 cm, if an arc subtends an angle of 60° at the centre of the circle, then the length of the arc is :

Mark practised

(A) 11 cm
(B) 44 cm
(C) \(\frac{22}{7}\) cm
(D) 22 cm

Marks: 1
Level: Basic
Years: 2024
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2024 Mathematics Basic, Set 430/4/1, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 22 cm, because length of arc = 60/360 × 2 × 22/7 × 21 = 22 cm.
Why? By the formula for the length of an arc of a sector, the arc is the angle fraction θ/360 of the whole circumference 2πr; applying it here gives the answer.

The full 1 mark is for the correct option.

Answer:

(D) 22 cm

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/4/1: option (D).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the sector-area formula 60/360 × 22/7 × 21² (231 cm²) instead of the arc-length formula; the question asks for length, not area.
Using θ/360 × πr (one radius instead of the circumference 2πr), which gives 11 cm, option A.
Forgetting the factor 60/360 and writing the full circumference 2 × 22/7 × 21 as the length of the 60° arc.

Q15. The area of the sector of a circle with radius 6 cm which subtends an angle of 60° at the centre of the circle is :

Mark practised

(A) \(\frac{142}{7}\) cm\(^{2}\)
(B) \(\frac{152}{7}\) cm\(^{2}\)
(C) \(\frac{132}{7}\) cm\(^{2}\)
(D) \(\frac{122}{7}\) cm\(^{2}\)

Marks: 1
Level: Basic
Years: 2024
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2024 Mathematics Basic, Set 430/4/3, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 132/7 cm², because (60/360) × (22/7) × 6² = 132/7 cm².
Why? The area of a sector is the fraction θ/360 of the whole circle's area, πr², and this calculation gives the option's value.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/4/3: option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using a decimal approximation for π instead of 22/7, so the answer does not match the given options.
Using the arc length formula (60/360) × 2πr, which gives a length, not an area.
Taking the diameter 12 cm as the radius instead of 6 cm.
Leaving the answer as 6π cm² without converting it to 132/7 cm², so it does not match the options.

Why this works

The sector is the part of the circular region enclosed by two radii and the corresponding arc. The whole circle is a sector of angle 360°, so the area for an angle of 1° is πr²/360. For an angle of 60°, the area is (60/360) × πr². With radius 6 cm and π = 22/7, this becomes (60/360) × (22/7) × 36 cm². Since 60/360 = 1/6 and (1/6) × 36 = 6, the area is 6 × 22/7 = 132/7 cm². Quick check: the full circle has area (22/7) × 36 cm², and 60/360 of that is 132/7 cm².

Key idea to remember

Area of a sector of a circle: see Key ideas used on this page.

Explanation

A sector is the part of a circular region enclosed by two radii and the corresponding arc. Its angle at the centre is θ. The whole circle has angle 360°, so the sector's area is the fraction θ/360 of the whole circle's area, πr². For this question, the angle at the centre is 60° and the radius is 6 cm, so the area is (60/360) × (22/7) × 36 = 132/7 cm².

Important formulas

Area of the sector of angle θ = (θ/360) × πr² · θ is the angle at the centre in degrees and r is the radius of the circle.
Area of a circle = πr² · The whole circle is a sector of angle 360°.

Q16. If the area of a sector of a circle is \(\frac{7}{20}\) of the area of the circle, then the angle at the centre is equal to

Mark practised

(A) \(110^\circ\)
(B) \(130^\circ\)
(C) \(100^\circ\)
(D) \(126^\circ\)

Marks: 1
Level: Standard
Years: 2024
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2024 Mathematics Standard, Set 30/3/2, Q6, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 126°, because the area of a sector of angle θ is (θ/360) × πr², so θ = (7/20) × 360 = 126°.
Why? By the formula for the area of a sector of angle θ, the sector area is (θ/360) times the circle's area, so this fraction is set equal to 7/20 and solved for θ.

The full 1 mark is for the correct option.

Answer:

(D) 126°

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/3/2: option (D).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Finding 7/20 of 180° instead of 360°, because the sector is compared with the whole circle, whose angle at the centre is 360°.
Using θ/100 in place of θ/360, so solving θ/100 = 7/20 gives θ = 35.
Inverting the fraction 7/20 to 20/7 before multiplying by 360, which makes the angle greater than 360°.

Why this works

The whole circle is a sector of angle 360°, so any sector of angle θ covers the fraction θ/360 of the circle's area. The question says this fraction is 7/20. Hence θ/360 = 7/20, so θ = (7/20) × 360 = 126. Quick check: 126/360 = 0.35 and 7/20 = 0.35, so the answer matches the given fraction.

Key idea to remember

Area of a sector of a circle: see Key ideas used on this page.

Explanation

For a circle of radius r, the area of the whole circle is πr². A sector with angle θ at the centre covers the fraction θ/360 of the circle. Therefore its area is (θ/360) × πr². This solution equates that fraction to 7/20 and solves for θ.

Important formulas

Area of the sector of angle θ = (θ/360) × πr² · Here r is the radius of the circle and θ is the angle of the sector in degrees.

Q17. The area of the sector of a circle of radius 12 cm is \(60\pi\text{ cm}^2\) . The central angle of this sector is :

Mark practised

(A) \(120^\circ\)
(B) \(6^\circ\)
(C) \(75^\circ\)
(D) \(150^\circ\)

Marks: 1
Level: Standard
Years: 2024
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2024 Mathematics Standard, Set 30/4/1, Q6, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 150°, because θ/360 × π(12)² = 60π gives θ = 360 × 60/144 = 150°.
Why? The area of a sector is the fraction θ/360 of the whole circle's area πr². Substituting the given radius and area and solving for θ gives the central angle.

The full 1 mark is for the correct option.

Answer:

(D) 150°

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/4/1: option (D).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Dividing the given area by the whole circle's area but forgetting to multiply by 360, so the angle comes out far too small.
Using the circumference formula 2πr in place of the area formula πr², which gives an angle far outside the options.

Why this works

A sector is the part of the circle enclosed by two radii and the arc between them. Its area is the fraction θ/360 of the whole circle's area πr². Here the whole circle has area π × 12² = 144π cm². The given sector area is 60π cm², so the sector is 60/144 of the circle. Therefore θ = 360 × 60/144 = 150°. Quick check: 150/360 of 144π is 60π cm², which matches the given area.

Key idea to remember

Area of a Sector of a Circle: see Key ideas used on this page.

Explanation

A sector of angle θ covers the fraction θ/360 of the whole circle, so its area is (θ/360) × πr². In this question, we substitute the given radius and area into this formula and solve the resulting equation for θ.

Important formulas

Area of the sector of angle θ = (θ/360) × πr² · θ is the angle of the sector in degrees and r is the radius of the circle.

Q18. The degree measure of the angle at the centre of a semicircle is :

Mark practised

(A) \(90^\circ\)
(B) \(60^\circ\)
(C) \(120^\circ\)
(D) \(180^\circ\)

Marks: 1
Level: Basic
Compartment: 2025
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2025 Compartment Mathematics Basic, Set 430/S/2, Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 180°, because a semicircle is half of a full circle, whose angle at the centre is 360°.
Why? The whole circle makes one complete turn of 360° at the centre; a semicircle is half of that turn, so it makes 180°.

The full 1 mark is for the correct option.

Answer:

(D) 180°

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 430/S/2: option (D).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 90° because a semicircle may look like a quarter turn, but a quarter of a circle is 90°, while a semicircle is half of 360°.
Choosing 120° because three equal sectors of a circle are 120° each, but a semicircle is not made of three such sectors.

Q19. The degree measure of the angle at the centre of a quadrant of a circle is :

Mark practised

(A) \(30^\circ\)
(B) \(60^\circ\)
(C) \(90^\circ\)
(D) \(180^\circ\)

Marks: 1
Level: Basic
Compartment: 2025
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2025 Compartment Mathematics Basic, Set 430/S/3, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 90°, because a quadrant is one-fourth of a circle and 1/4 × 360° = 90°.
Why? A full circle makes an angle of 360° at the centre, so one-fourth of it has angle 360° ÷ 4.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 430/S/3: option (C).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 180° because 180° is half of 360°, but a quadrant is one-fourth, not one-half, of the circle.
Choosing 60° because a quadrant sounds like a quarter and 360 ÷ 6 = 60; the correct division is by 4.

Q20. The difference between the areas of a semicircle of diameter 2r units and the largest triangle drawn inside the semicircle is :

Mark practised

(A) \(\frac{4r^2}{7}\) sq. units
(B) \(\frac{15r^2}{7}\) sq. units
(C) \(\frac{15r^2}{14}\) sq. units
(D) \(\frac{37r^2}{14}\) sq. units

Marks: 1
Level: Standard
Compartment: 2025
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2025 Compartment Mathematics Standard, Set 30/S/2, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 4r²/7 sq. units, because the semicircle area is 1/2 × πr² and the largest triangle has base 2r and height r, so with π = 22/7 the difference is 4r²/7.
Why? A semicircle is half of the circular region, and the greatest height from the diameter to the arc is the radius, so the largest triangle has base 2r and height r.

The full 1 mark is for the correct option.

Answer:

(A) 4r²/7 sq. units

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 30/S/2: option (A).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Taking the semicircle's area as πr² instead of 1/2 πr², which gives option (B) 15r²/7.
Taking the triangle's base as r instead of 2r, which gives option (C) 15r²/14.
Writing the final answer as 4/7 without r², because every area here is in square units.

Why this works

The diameter of length 2r is the longest chord inside the semicircle, and the point of the semicircle farthest from the diameter is at height r. So the largest triangle has area 1/2 × 2r × r = r². A semicircle is a sector of angle 180°, so its area is 180/360 × πr² = 1/2 πr². With π = 22/7, the difference is 1/2 × 22/7 r² − r² = 4r²/7, which is option (A). Quick check with r = 7: semicircle area = 77, triangle area = 49, difference = 28, and 4/7 × 49 = 28.

Key idea to remember

Area of a sector of a circle: see Key ideas used on this page.

Explanation

A semicircle is a sector with angle 180°, so its area is half the circle's area. The largest triangle inside it uses the diameter as base and the radius as height, so its area is r². Using π = 22/7, the difference is 4r²/7.

Important formulas

Area of the sector of angle θ = θ/360 × πr² · A semicircle has θ = 180°, so its area is 1/2 πr².
Area of a triangle = 1/2 × base × height · For the largest triangle in the semicircle, base = 2r and height = r, giving area r².

Q21. A and B are sectors of two different circles. Radius of sector A is double of that of sector B whereas central angle of sector B is double the central angle of sector A. The ratio of the area of sector A to the area of sector B is :

Mark practised

(A) 1 : 1
(B) 1 : 2
(C) 2 : 1
(D) 4 : 1

Marks: 1
Level: Standard
Compartment: 2025
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2025 Compartment Mathematics Standard, Set 30/S/3, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 2 : 1, because Area A : Area B = θ(2r)² : (2θ)r², which simplifies to 4 : 2, that is 2 : 1.
Why? Area of a sector = (θ/360) × πr², so two sectors are compared by θr²; the common factor (θ/360) × π cancels in a ratio.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 30/S/3: option (C).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing option A (1 : 1), because the radius of A is doubled without squaring it, so the factor 4 is missed.
Choosing option B (1 : 2), because the ratio is reversed and written as B : A instead of A : B.
Choosing option D (4 : 1), because the doubled central angle of sector B is ignored.

Why this works

Let the radius of sector B be r and the central angle of sector A be θ. Then sector A has radius 2r and sector B has central angle 2θ. Using the formula for area of a sector, Area A = (θ/360) × π(2r)² and Area B = (2θ/360) × πr². The common factor (θ/360) × π cancels, so Area A : Area B = (2r)² : 2r². Since (2r)² = 4r², the ratio is 4r² : 2r², that is 2 : 1.

Key idea to remember

Area of a Sector of a Circle: see Key ideas used on this page.

Explanation

A sector is the part of the circular region enclosed by two radii and the corresponding arc. Its area is (θ/360) × πr². This solution writes both areas with this formula and cancels the common factor (θ/360) × π; the comparison then depends only on (central angle) × (radius)².

Important formulas

Area of a sector of angle θ = (θ/360) × πr² · θ is the central angle in degrees and r is the radius of that circle.

Q22. If an arc of a circle of diameter 10 cm subtends an angle of 144° at the centre of the circle, then the length of the arc is :

Mark practised

(A) 2π cm
(B) 4π cm
(C) 5π cm
(D) 6π cm

Marks: 1
Level: Standard
Years: 2025
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2025 Mathematics Standard, Set 30/2/3, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 4π cm, because the radius is 5 cm and the length of the arc = (144/360) × 2π × 5 = 4π cm.
Why? The whole circumference 2πr corresponds to 360° at the centre, so the arc's fraction of the circumference equals θ/360.

The full 1 mark is for the correct option.

Answer:

(B) 4π cm

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/2/3: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the diameter 10 cm as the radius, which gives 8π cm instead of 4π cm.
Using the sector area formula (θ/360) × πr², which gives 10π cm², not the arc length.
Thinking 144° is one-fourth of 360°, which would give 5π/2 cm instead of 4π cm.

Q23. The perimeter of sector of a circle of radius 21 cm and central angle \(60^\circ\), is

Mark practised

(A) 22 cm
(B) 44 cm
(C) 64 cm
(D) 273 cm

Marks: 1
Level: Standard
Years: 2026
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2026 Mathematics Standard, Set 30(B), Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 64 cm, because the perimeter of the sector = length of the arc + 2 × radius = (60/360 × 2 × 22/7 × 21) + 2 × 21 = 22 + 42 = 64 cm.
Why? The perimeter of a sector is the sum of its curved boundary (the arc) and the two radii, and the NCERT formula for the length of an arc of angle θ is θ/360 × 2πr.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30(B): option (C).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Giving 22 cm, the arc length alone, and forgetting the two radii that also form the boundary of the sector.
Adding only one radius to the arc length, getting 43 cm, instead of adding both 21 cm and 21 cm.
Using the area formula 60/360 × 22/7 × 21², which gives 231 cm², instead of finding the perimeter.

Why this works

A sector is the region enclosed by two radii and the corresponding arc, so its perimeter has three parts: the arc and the two straight radii. For a central angle of 60°, the arc is 60/360 of the full circle, that is, one-sixth. The circumference is 2 × 22/7 × 21 cm = 132 cm, so the arc length is 132/6 cm = 22 cm. The two radii together are 21 + 21 = 42 cm. Therefore the perimeter is 22 + 42 = 64 cm. Quick check: 64 − 42 = 22, and 6 × 22 = 132, which matches the circumference, so the arc is indeed one-sixth of the circle.

Key idea to remember

Length of an arc of a sector of a circle: see Key ideas used on this page.

Explanation

The NCERT text derives the arc length by the unitary method: a full circle has angle 360° and circumference 2πr, so an angle of θ° gives θ/360 of the circumference, i.e. θ/360 × 2πr. A sector's boundary is this arc plus the two radii, so its perimeter is θ/360 × 2πr + 2r. Using the given angle and radius in this formula gives the working in the exam answer.

Important formulas

Length of an arc = θ/360 × 2πr · θ is the angle of the sector in degrees and r is the radius.
Perimeter of a sector = θ/360 × 2πr + 2r · Add the two radii to the arc length; the NCERT text gives the arc-length part.

Q24. The area (in \(\text{cm}^2\)) of a sector of a circle of radius 14 cm cut off by an arc of length 22 cm is :

Mark practised

(A) 77
(B) 308
(C) 154
(D) 462

Marks: 1
Level: Basic
Years: 2026
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2026 Mathematics Basic, Set 430(B), Q14, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 154, because area of the sector = (arc length × radius)/2 = (22 × 14)/2 = 154 cm².
Why? A sector's area and its arc length both use the same fraction θ/360 of the whole circle, so the area is half the product of the arc length and the radius.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2026 Main, Paper 430(B): option (C).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 308, which is arc length × radius without halving.
After finding the sector fraction, using the radius 14 instead of r² in πr².

Why this works

Let θ be the angle of the sector. The circumference of the circle is 2 × (22/7) × 14 = 88 cm. The arc of length 22 cm is one-fourth of the circumference, so θ/360 = 22/88 = 1/4. Therefore the sector area is one-fourth of the whole circle: (1/4) × (22/7) × 14² = 154 cm². This matches the shorter formula area = (arc length × radius)/2.

Key idea to remember

Area of a Sector of a Circle: see Key ideas used on this page.

Explanation

NCERT gives the area of a sector as (θ/360) × πr² and the length of an arc as (θ/360) × 2πr. Both contain the same fraction θ/360, so when the arc length and radius are known, the sector area can be written as (arc length × radius)/2.

Important formulas

Length of an arc of a sector = (θ/360) × 2πr · Use this when the arc length and radius are given.
Area of a sector = (θ/360) × πr² · This is the NCERT area formula for a sector.
Area of a sector = (arc length × radius)/2 · Combining the two NCERT formulas removes θ.

Q25. The minute hand of a clock is 21 cm long. The distance covered by the tip of minute hand from 2:10 pm to 2:25 pm is :

Mark practised

(A) 346.5 cm
(B) 33 cm
(C) 66 cm
(D) 16.5 cm

Marks: 1
Level: Basic
Years: 2026
Topic: 11.1 Areas of Sector and Segment of a Circle
Source: (CBSE 2026 Mathematics Basic, Set 430(B), Q15, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 33 cm, because the tip turns through 90° in 15 minutes, so arc length = (90/360) × 2 × (22/7) × 21 = 33 cm.
Why? The tip of the minute hand moves along a circular arc, and the length of an arc is a fraction of the whole circumference 2πr.

The full 1 mark is for the correct option.

Answer:

(B) 33 cm

✓ Answer matches the CBSE answer key, 2026 Main, Paper 430(B): option (B).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 346.5 cm, which is the area of the sector (90/360 × 22/7 × 21²), not the distance covered.
Taking the time as half an hour instead of 15 minutes gives 66 cm.
Taking 21 cm as the diameter (radius 10.5 cm) gives 16.5 cm.

Why this works

From 2:10 to 2:25 the time is 15 minutes. In 60 minutes the minute hand turns 360°, so in 15 minutes it turns (15/60) × 360 = 90°. The distance covered by the tip is the length of the arc it traces. Length of arc = (90/360) × 2 × (22/7) × 21 = 33 cm. As a quick check, 15 minutes is one quarter of an hour, so the arc is one quarter of the circumference; the circumference is 2 × (22/7) × 21 = 132 cm, and 132 ÷ 4 = 33 cm. Another correct method also earns full marks.

Key idea to remember

Length of an arc of a sector of a circle: see Key ideas used on this page.

Explanation

The whole circumference 2πr belongs to an angle of 360° at the centre. So an angle of θ degrees gives the fraction θ/360 of the circumference. Here the minute hand is the radius of the circle, and it turns through 90° in 15 minutes.

Important formulas

Length of an arc = (θ/360) × 2πr · θ is the angle of the sector in degrees and r is the radius.

1 / 25

Key ideas used on this page

Length of an arc of a sector of a circle
Used in questions 2 3 6 7 9 10 12 14 23 25
Area of a sector of a circle
Used in questions 1 15 16 17 20 21 24
Length of an arc of a sector of a circle and area of a sector of a circle
Used in question 4
Segment of a circle
Used in question 5
Angle of a sector
Used in question 8
Unitary Method
Used in question 11
Circumference and area of a circle
Used in question 13
Angle at the centre
Used in question 18
Sector of a circle
Used in question 19
Length of an arc of a sector
Used in question 22

How to use these questions

Try each question on paper first. Then read the steps: the mark beside each step shows what the examiner looks for. If a step is new to you, revise Section 11.1 of the NCERT chapter. For the exam pattern and the repeated questions of the whole chapter, read the Areas Related to Circles chapter overview.

To revise the lessons behind these questions, read the NCERT chapter, exactly as NCERT publishes it on its own website.

Official papers behind this page

Every question on this page comes from an official CBSE Class 10 Mathematics board paper. The board publishes the question papers on cbse.gov.in and the marking schemes on cbse.gov.in, where you can see how marks are given.

Year	Exam	Course	Question paper	Marking scheme
2026	Main	Standard (041)	Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper.	Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 22.8 MB, Class 10 Maths Standard (041)) to see how marks are given.
2026	Main	Basic (241)	Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper.	Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 29.1 MB, Class 10 Maths Basic (241)) to see how marks are given.
2025	Main	Standard (041)	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 54.5 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Main	Basic (241)	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 49.9 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Compartment	Standard (041)	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 3.4 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 8.1 MB, Class 10 Maths Standard (041)) to see how marks are given.
2025	Compartment	Basic (241)	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 2.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 18.7 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Main	Standard (041)	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 35.3 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2024	Main	Basic (241)	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 43.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.8 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Compartment	Basic (241)	Open CBSE's official question paper for the 2024 compartment exam (one ZIP file of about 2.2 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 compartment exam (one ZIP file of about 5.6 MB, Class 10 Maths Basic (241)) to see how marks are given.
2023	Main	Standard (041)	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 44 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 17.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2023	Main	Basic (241)	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 49.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 26 MB, Class 10 Maths Basic (241)) to see how marks are given.
2023	Compartment	Standard (041)	Open CBSE's official question paper for the 2023 compartment exam (one ZIP file of about 7.6 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 compartment exam (one ZIP file of about 3.3 MB, Class 10 Maths Standard (041)) to see how marks are given.

Sources and session

Each question is reproduced as printed in its CBSE paper, with its year, paper code and question number; only the layout is tidied. The same question printed in several paper sets counts once, and the card says how often it was printed in 2023–2026. These counts describe past papers; they are not predictions.

Every solution is original and follows NCERT Mathematics Class X (reprint 2026-27). The mark beside each step follows the official CBSE marking scheme where the scheme shows it; a mark labelled “(suggested)” follows the board’s usual pattern for that type of question.

Reference: CBSE Class 10 Mathematics board question papers via cbse.gov.in; all solutions are original. learncbse.net is an independent website and is not affiliated with CBSE or NCERT.

1-mark questions on Areas Related to Circles

All 25 Areas Related to Circles 1-mark questions

Q1. The angle of the sector of a circle whose area is one-eighth of the area of the circle is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q2. OACB is a quadrant of a circle with centre O and radius 7 cm where ACB is the arc. Then the perimeter of the quadrant is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q3. What is the length of the arc of the sector of a circle with radius 14 cm and of central angle 90° ?

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q4. If length of an arc of a circle subtending an angle \(\theta\) at the centre is numerically equal to the area of the sector formed by it, then the radius of the circle is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q5. The region between a chord and either of the two arcs of a circle is called :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q6. The perimeter of the sector of a circle of radius 21 cm which subtends an angle of \(60^\circ\) at the centre of circle, is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q7. A piece of wire 20 cm long is bent into the form of an arc of a circle of radius 60/π cm. The angle subtended by the arc at the centre of the circle is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q8. If a large circular pizza is divided into 5 equal sectors, then the central angle of each sector will be :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q9. The length of arc subtending an angle of \(210^\circ\) at the centre of the circle, is \(\frac{44}{3}\) cm. The radius of the circle is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q10. The length of the arc of the sector of a circle with radius 21 cm and of central angle \(60^\circ\) , is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q11. The hour hand of a clock is 7 cm long. The angle swept by it between 7:00 a.m. and 8:10 a.m. is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q12. What is the length of arc of a circle of radius 7 cm which subtends an angle of \(90^\circ\) at the centre of the circle ?

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q13. If the perimeter and the area of a circle are numerically equal, then the radius of the circle is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q14. In a circle of radius 21 cm, if an arc subtends an angle of 60° at the centre of the circle, then the length of the arc is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q15. The area of the sector of a circle with radius 6 cm which subtends an angle of 60° at the centre of the circle is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q16. If the area of a sector of a circle is \(\frac{7}{20}\) of the area of the circle, then the angle at the centre is equal to

Step-by-step solution

Step 1: Choose the option