Areas Related to Circles CBQs — CBSE Class 10 (2027) with step-wise solutions

Sample Areas Related to Circles CBQs

1. Question 1MCQ1 mark
   The minute hand of a wall clock in a school library is 21 cm long. The librarian notices the time as 7:15 a.m. when she opens the library and 7:35 a.m. when the first student walks in. (Use π = 22/7) What area of the clock face is swept by the minute hand between these two times?
   • 462 cm²
   • 231 cm²
   • 44 cm²
   • 1386 cm²
   Show step-wise solution

   1. [1 mark] In 20 minutes the minute hand turns 20 × 6° = 120°, so area swept = (120/360) × (22/7) × 21 × 21 = (1/3) × 1386 = 462 cm².

   Answer: 462 cm²

2. Question 2MCQ1 mark
   A curved ramp on a highway flyover is built as an arc of a circle of radius 105 m. While driving along the ramp, a car turns through an angle of 60° about the centre of that circle. (Use π = 22/7) What is the length of the curved ramp?
   • 220 m
   • 110 m
   • 55 m
   • 660 m
   Show step-wise solution

   1. [1 mark] Arc length = (60/360) × 2 × (22/7) × 105 = (1/6) × 660 = 110 m.

   Answer: 110 m

3. Question 3Assertion-Reason1 mark
   The minute hand of a railway station clock is 14 cm long. (Use π = 22/7) Assertion (A): In 30 minutes, the tip of the minute hand travels a distance of 44 cm. Reason (R): In 30 minutes the minute hand turns through 180°, and the length of an arc of angle θ in a circle of radius r is (θ/360) × 2πr.
   • (A) Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
   • (B) Both Assertion and Reason are true but Reason is NOT the correct explanation of Assertion.
   • (C) Assertion is true but Reason is false.
   • (D) Assertion is false but Reason is true.
   Show step-wise solution

   1. [1 mark] (A): Arc = (180/360) × 2 × (22/7) × 14 = 44 cm, so A is true; R is true and is exactly the reasoning used to get 44 cm.

   Answer: (A)

4. Question 4Short answer2 marks
   On a hill section, a railway track runs along an arc of a circle of radius 630 m. While going round this curve, the direction of the train changes by 40°, so the curved track subtends an angle of 40° at the centre of the circle. Find the length of the curved part of the track. (Use π = 22/7.)
   Show step-wise solution

   1. [1 mark] Length of arc = (θ/360°) × 2πr = (40/360) × 2 × (22/7) × 630 = (1/9) × 3960.
   2. [1 mark] Length of the curved track = 440 m.

   Answer: 440 m

5. Question 5Case study4 marks
   A lighthouse stands on a rocky headland. Its beam is visible up to 14 km out at sea, but the land blocks it on one side, so the beam lights only a sector of the sea of angle 90°, with its centre at the lighthouse. (Take π = 22/7.) Based on the above information, answer the following questions: (i) Find the length of the outer curved edge of the region of sea lit by the beam. [1 mark] (ii) Find the area of sea lit by the beam. [1 mark] (iii) A stronger lamp is fitted, so the beam is now visible up to 21 km over the same 90° sector. Find the increase in the area of sea lit. [2 marks]
   Show step-wise solution

   1. [1 mark] (i) Arc length = (90/360) × 2πr = (1/4) × 2 × (22/7) × 14 = 22 km.
   2. [1 mark] (ii) Area = (90/360) × πr² = (1/4) × (22/7) × 14 × 14 = 154 km².
   3. [1 mark] (iii) New area = (1/4) × (22/7) × 21 × 21 = 346.5 km².
   4. [1 mark] (iii) Increase in area = 346.5 − 154 = 192.5 km².

   Answer: (i) 22 km (ii) 154 km² (iii) 192.5 km²

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