Class 10 Mathematics: Areas Related to Circles important questions
Areas Related to Circles is a chapter of Class 10 Mathematics for the CBSE. These 17 questions are marked Very Important, Important or Practice, and each answer has been checked by our team. Use them to revise, not as a prediction of what will come in the exam.
Practice · 1 marks
1. The area of the circle that can be inscribed in a square of side 6 cm is: (a) 36π cm² (b) 18π cm² (c) 12π cm² (d) 9π cm²
(d) 9π cm² The radius is 3 cm.
Practice · 1 marks
2. The number of revolutions made by a circular wheel of radius 0.25 m in rolling a distance of 11 km is: (a) 2800 (b) 4000 (c) 5500 (d) 7000
(d) 7000 Circumference = 2 × (22/7) × 0.25 = 11/7 m. Revolutions = 11000 ÷ (11/7) = 7000.
Important · 2 marks
3. The minute hand of a clock is 6 cm long. Find the area swept by it from 7:05 p.m. to 7:40 p.m.
35 minutes × 6° = 210°. Area = (210/360) × (22/7) × 6 × 6 = 66 cm².
Important · 2 marks
4. Arcs of radius 7 cm are drawn with the vertices A, B, C and D of a quadrilateral ABCD as centres, inside the quadrilateral. Find the total area of the four sectors.
The angles of a quadrilateral add up to 360°, so the four sectors make one full circle. Area = (22/7) × 7 × 7 = 154 cm².
Practice · 1 marks
5. If the perimeter and the area of a circle are numerically equal, the radius is: (a) 2 units (b) π units (c) 4 units (d) 7 units
(a) 2 units 2πr = πr² gives r = 2.
Practice · 1 marks
6. A new circular park is to equal in area the sum of two circular parks of diameters 16 m and 12 m. The radius of the new park is: (a) 10 m (b) 15 m (c) 20 m (d) 24 m
(a) 10 m πR² = π(8²) + π(6²) = 100π, so R = 10 m.
Important · 2 marks
7. With the vertices A, B and C of ΔABC as centres, arcs of radius 14 cm are drawn, and the three sectors so formed are removed from the triangle. Find the total area removed.
The three angles add up to 180°, so the removed area = (180/360) × (22/7) × 14 × 14 = 308 cm².
Practice · 1 marks
8. Assertion (A): If the radius of a sector of a circle is reduced to half and its angle is doubled, then the perimeter of the sector remains the same. Reason (R): The length of the arc subtending angle θ at the centre of a circle of radius r is πrθ/180. A. Both A and R are true, and R is the correct explanation of A. B. Both A and R are true, and R is not the correct explanation of A. C. A is true but R is false. D. A is false but R is true.
D. A is false but R is true. The arc length stays the same (πrθ/180), but the two radii are halved, so the perimeter changes.
Important · 3 marks
9. The minute hand of a wall clock is 18 cm long. Find the area of the face of the clock swept by the minute hand in 35 minutes.
The minute hand turns 6° per minute, so 35 minutes = 210°. Area = (210/360) × (22/7) × 18 × 18 = 594 cm².
Important · 3 marks
10. AB is a chord of a circle with centre O such that ∠AOB = 60°. If OA = 14 cm, find the area of the minor segment. (Take √3 = 1.73)
Area of sector = (60/360) × (22/7) × 14 × 14 ≈ 102.67 cm². ΔOAB is equilateral: area = (√3/4) × 14² = 84.77 cm². Minor segment = 102.67 − 84.77 ≈ 17.89 cm².
Practice · 1 marks
11. The area of the circle inscribed in a square of side 6 cm is: (A) 36π cm² (B) 18π cm² (C) 12π cm² (D) 9π cm²
(D) 9π cm² The radius is 3 cm, so area = π × 3² = 9π cm².
Practice · 1 marks
12. The area (in cm²) of a sector of a circle of radius 21 cm cut off by an arc of length 22 cm is: (A) 441 (B) 321 (C) 231 (D) 221
(C) 231 Area = ½ × l × r = ½ × 22 × 21 = 231 cm².
Important · 2 marks
13. A horse, a cow and a goat are each tied with 14 m ropes at the corners A, B and C of a triangular grass field with sides 35 m, 40 m and 50 m. Find the area of grass they can graze.
Each grazes a sector of radius 14 m, and the three angles of the triangle add up to 180°. Total area = (180°/360°) × (22/7) × 14 × 14 = 308 m².
Important · 2 marks
14. Find the area of the major segment (in terms of π) of a circle of radius 5 cm formed by a chord subtending 90° at the centre.
Minor segment = sector − triangle = (90/360) × π × 25 − ½ × 5 × 5 = 25π/4 − 25/2. Major segment = 25π − (25π/4 − 25/2) = (75π/4 + 25/2) cm².
Practice · 1 marks
15. The diameter of a car wheel is 21 cm. The number of complete revolutions it makes in moving 66 km is: (A) 10⁴ (B) 10⁵ (C) 10⁶ (D) 10⁷
(B) 10⁵ Circumference = π × 21 = (22/7) × 21 = 66 cm. 66 km = 66 × 10⁵ cm, so revolutions = 66 × 10⁵ ÷ 66 = 10⁵.
Practice · 1 marks
16. An arc of a circle has length 6π cm and its sector has area 24π cm². The radius is: (A) 4 cm (B) 8 cm (C) 16 cm (D) 18 cm
(B) 8 cm Area of sector = ½ × arc length × r, so 24π = ½ × 6π × r and r = 8 cm.
Important · 2 marks
17. Find the area of a sector of a circle of diameter 28 cm if the length of its arc is 22 cm.
Radius = 14 cm. Area of sector = ½ × arc length × r = ½ × 22 × 14 = 154 cm².
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