Assertion (A): A wire is bent into a circle of radius 14 cm. If the same wire is rebent to form a sector of a circle of radius 14 cm with arc length 22 cm, then the angle of the sector is 90° (Use π = 22/7). Reason (R): The length of the wire remains unchanged, so the perimeter of the sector equals the circumference of the original circle.

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

In a circle of radius 21 cm, a chord AB subtends an angle of 60° at the centre O. Find (i) the area of the minor sector OAB, (ii) the area of the minor segment formed by the chord AB, (iii) the area of the major sector. (Use π = 22/7 and √3 = 1.73)

(i) 231 cm² (ii) 40.07 cm² (iii) 1155 cm²

A children's park is designed in the shape of a rectangle with semicircular ends. The rectangular portion has length 28 m. The total perimeter of the park is 100 m. (Use π = 22/7) (1) Find the radius of the semicircular ends. (2) Find the area of the park. (3) If the park is to be levelled at a cost of Rs 25 per square metre, calculate the total levelling cost.

(i) 7 m (ii) 546 m^2 (iii) Rs 13650

Class 10 Maths Chapter 11: Areas Related to Circles — Important Questions & Sample Paper

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Chapter 11 of CBSE Class 10 Mathematics, 'Areas Related to Circles,' builds upon students' knowledge of circles and perimeter/area formulas from earlier classes. This chapter focuses on computing the area and perimeter of circular regions beyond full circles. Key concepts include the area and perimeter of a semicircle, the area of a quadrant, the area of a sector (using the proportion of the central angle to 360°), and the length of an arc. Students also learn to calculate the area of a segment by subtracting the area of a triangle from the area of the corresponding sector. Real-world applications are emphasized through problems involving clocks, such as finding the area swept by minute or hour hands over a time interval. Typical exam questions ask for the area of a sector given radius and angle, the area of a minor or major segment, the perimeter of a semicircular protractor, or the area covered by a pendulum or clock hand. Mastery of these formulas and the ability to convert between degrees and proportions are essential. This chapter not only sharpens geometry skills but also reinforces algebraic manipulation, as many problems require substituting π = 22/7 or 3.14 and simplifying expressions. Practice with varied problems helps students understand the practical significance of circular measures.

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Mathematics — Areas Related to Circles

Class 10Time: 3 hrsMax Marks: 80

SECTION A

1.
A chord of a circle of radius 10 cm subtends an angle of 120° at the centre. The area of the minor segment (π = 3.14, √3 = 1.732) is:
(a) 61.4 cm²(b) 62.8 cm²(c) 59.2 cm²(d) 64.6 cm²
1
2.
The minute hand of a clock is 12 cm long. The area swept by it from 12:00 noon to 12:20 p.m. (π = 3.14) is:
(a) 75.36 cm²(b) 113.04 cm²(c) 150.72 cm²(d) 37.68 cm²
1
3.
If the radius of a circle is doubled, then its area becomes:
(a) 2 times(b) 4 times(c) 8 times(d) unchanged
1

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In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question type	Marks each	In our bank
Multiple Choice (MCQ)	1 mark	13
Assertion–Reason	1 mark	6
Short Answer	2 marks	8
Short Answer	3 marks	6
Long Answer	5 marks	5
Case Study	4 marks	6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

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Q1. A chord of a circle of radius 10 cm subtends an angle of 120° at the centre. The area of the minor segment (π = 3.14, √3 = 1.732) is:
1 mark
Multiple Choice (MCQ)
(A) 61.4 cm²(B) 62.8 cm²(C) 59.2 cm²(D) 64.6 cm²
▸ Answer▾ Answer
61.4 cm²
Q2. The minute hand of a clock is 12 cm long. The area swept by it from 12:00 noon to 12:20 p.m. (π = 3.14) is:
1 mark
Multiple Choice (MCQ)
(A) 75.36 cm²(B) 113.04 cm²(C) 150.72 cm²(D) 37.68 cm²
▸ Answer▾ Answer
150.72 cm²
Q3. If the radius of a circle is doubled, then its area becomes:
1 mark
Multiple Choice (MCQ)
(A) 2 times(B) 4 times(C) 8 times(D) unchanged
▸ Answer▾ Answer
4 times
Q4. A circle has a radius of 14 cm. Taking π = 22/7, its circumference is:
1 mark
Multiple Choice (MCQ)
(A) 44 cm(B) 88 cm(C) 132 cm(D) 176 cm
▸ Answer▾ Answer
88 cm
Q5. Assertion (A): A wire is bent into a circle of radius 14 cm. If the same wire is rebent to form a sector of a circle of radius 14 cm with arc length 22 cm, then the angle of the sector is 90° (Use π = 22/7). Reason (R): The length of the wire remains unchanged, so the perimeter of the sector equals the circumference of the original circle.
1 mark
Assertion–Reason
(A) Both A and R are true and R is the correct explanation of A.(B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
▸ Answer▾ Answer
A is false but R is true.
Q6. Find the area of a quadrant of a circle if its radius is 14 cm. (Use π = 22/7)
2 marks
Short Answer
▸ Answer▾ Answer
154 cm²
Q7. A round table cloth of radius 28 cm has a pattern of a regular hexagon inscribed in it. The design covers the region outside the hexagon but inside the circle. Find the total area of the design. (Take π = 22/7 and √3 = 1.73)
2 marks
Short Answer
▸ Answer▾ Answer
429.52 cm² (approx.)
Q8. A sector of a circle of radius 7 cm has an area of 77 cm². Find the length of the corresponding arc. (Use π = 22/7)
3 marks
Short Answer
▸ Answer▾ Answer
22 cm
Q9. A square is inscribed in a circle of radius 7 cm. Find the area of the region inside the circle but outside the square. (Use π = 22/7)
3 marks
Short Answer
▸ Answer▾ Answer
56 cm²
Q10. In a circle of radius 21 cm, a chord AB subtends an angle of 60° at the centre O. Find (i) the area of the minor sector OAB, (ii) the area of the minor segment formed by the chord AB, (iii) the area of the major sector. (Use π = 22/7 and √3 = 1.73)
5 marks
Long Answer
▸ Answer▾ Answer
(i) 231 cm² (ii) 40.07 cm² (iii) 1155 cm²
Q11. Three horses are tethered at the three corners of a triangular field with sides measuring 20 m, 30 m, and 40 m. Each horse is tied with a 7 m long rope. Assuming the horses can graze only within the field, compute the total area over which the horses can graze. (Use π = 22/7)
5 marks
Long Answer
▸ Answer▾ Answer
77 m²
Q12. A children's park is designed in the shape of a rectangle with semicircular ends. The rectangular portion has length 28 m. The total perimeter of the park is 100 m. (Use π = 22/7)
4 marks
Case Study
1. (i) Find the radius of the semicircular ends.1 mark
2. (ii) Find the area of the park.2 marks
3. (iii) If the park is to be levelled at a cost of Rs 25 per square metre, calculate the total levelling cost.1 mark
▸ Answer▾ Answer
(i) 7 m (ii) 546 m^2 (iii) Rs 13650

Frequently asked questions

What are the key formulas in Class 10 Areas Related to Circles?

Key formulas include: circumference = 2πr, area of circle = πr², area of semicircle = πr²/2, perimeter of semicircle = πr + 2r, area of quadrant = πr²/4, arc length = (θ/360) × 2πr, area of sector = (θ/360) × πr², area of segment = area of sector – area of corresponding triangle.

How do you solve clock problems involving areas swept by hands?

Determine the angle swept by the minute hand (moves 6° per minute) or hour hand (moves 0.5° per minute). Convert the given time into an angle, then use the area of a sector formula (θ/360) × πr² with the hand length as the radius.

What is the difference between a sector and a segment of a circle?

A sector is the region between two radii and the arc. A segment is the region between a chord and the arc. The minor segment is the sector minus the triangle formed by the radii and chord.

How do I find the area of a major segment?

The area of the major segment is the area of the full circle minus the area of the corresponding minor segment. Alternatively, it is the area of the major sector plus the area of the triangle.

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