Areas Related to Circles — Class 10 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Areas Related to Circles" — 9 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Areas Related to Circles" — 9 important questions with detailed answers for CBSE b…
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NCERT Solutions for Class 10 Mathematics — Areas Related to Circles. Step-by-step answers to all 9 textbook questions from this chapter, written for CBSE board preparation and free to use.
Areas Related to Circles — All 9 Questions Solved
Q1. A sector of a circle has radius 7 cm and central angle 60°. Find the area of the sector. (Use π = 22/7)
Step 1: Area of a sector = (θ/360°) × πr²
Step 2: Here, θ = 60°, r = 7 cm, π = 22/7
Step 3: Area = (60°/360°) × (22/7) × 7²
Step 4: Area = (1/6) × (22/7) × 49
Step 5: Area = (1/6) × 22 × 7
Step 6: Area = (1/6) × 154
Step 7: Area = 154/6 = 77/3 cm²
Step 8: Area ≈ 25.67 cm²
Final Answer: 77/3 cm² or approximately 25.67 cm²
Q2. Find the perimeter of a sector with radius 10.5 cm and arc length 11 cm.
Step 1: Perimeter of a sector = 2r + arc length
Step 2: Here, r = 10.5 cm, arc length = 11 cm
Step 3: Perimeter = 2(10.5) + 11
Step 4: Perimeter = 21 + 11
Step 5: Perimeter = 32 cm
Final Answer: 32 cm
Q3. The radius of a circle is 14 cm. Find the area of a segment if the central angle of the corresponding sector is 90°. (Use π = 22/7)
Step 1: Area of segment = Area of sector − Area of triangle
Step 2: Area of sector = (θ/360°) × πr² = (90°/360°) × (22/7) × 14²
Step 3: Area of sector = (1/4) × (22/7) × 196
Step 4: Area of sector = (1/4) × 22 × 28
Step 5: Area of sector = (1/4) × 616 = 154 cm²
Step 6: Area of triangle = (1/2) × r² × sin(θ) = (1/2) × 14² × sin(90°)
Step 7: Area of triangle = (1/2) × 196 × 1 = 98 cm²
Step 8: Area of segment = 154 − 98 = 56 cm²
Final Answer: 56 cm²
Q4. A circle has two sectors with central angles 120° and 150°. If the radius is 6 cm, find the difference between their areas. (Use π = 22/7)
Step 1: Area of sector 1 = (120°/360°) × πr²
Step 2: Area of sector 1 = (1/3) × (22/7) × 6²
Step 3: Area of sector 1 = (1/3) × (22/7) × 36
Step 4: Area of sector 1 = (1/3) × 22 × 36/7
Step 5: Area of sector 1 = (22 × 12)/7 = 264/7 cm²
Step 6: Area of sector 2 = (150°/360°) × πr²
Step 7: Area of sector 2 = (5/12) × (22/7) × 36
Step 8: Area of sector 2 = (5/12) × 22 × 36/7
Step 9: Area of sector 2 = (5 × 22 × 3)/7 = 330/7 cm²
Step 10: Difference = 330/7 − 264/7 = 66/7 ≈ 9.43 cm²
Final Answer: 66/7 cm² or approximately 9.43 cm²
Q5. Find the area of a segment of a circle with radius 5 cm and chord length 6 cm.
Step 1: First, find the central angle using the chord length
Step 2: For a chord, chord = 2r × sin(θ/2)
Step 3: 6 = 2(5) × sin(θ/2)
Step 4: sin(θ/2) = 6/10 = 0.6
Step 5: θ/2 = 36.87°, so θ ≈ 73.74° ≈ 74° (approximately)
Step 6: Using cos formula: cos(θ/2) = √(1 − 0.36) = √0.64 = 0.8
Step 7: Area of sector = (θ/360°) × πr² = (74°/360°) × (22/7) × 25
Step 8: Area of sector ≈ 16.09 cm²
Step 9: Area of triangle = (1/2) × r² × sin(θ) = (1/2) × 25 × sin(74°)
Step 10: Area of triangle ≈ (1/2) × 25 × 0.96 ≈ 12 cm²
Step 11: Area of segment = 16.09 − 12 ≈ 4.09 cm²
Final Answer: Approximately 4.1 cm²
Q6. The length of an arc of a circle is 22 cm and the radius is 21 cm. Find the area of the sector. (Use π = 22/7)
Step 1: Arc length = (θ/360°) × 2πr
Step 2: 22 = (θ/360°) × 2 × (22/7) × 21
Step 3: 22 = (θ/360°) × 2 × 22 × 3
Step 4: 22 = (θ/360°) × 132
Step 5: θ/360° = 22/132 = 1/6
Step 6: θ = 60°
Step 7: Area of sector = (θ/360°) × πr² = (60°/360°) × (22/7) × 21²
Step 8: Area of sector = (1/6) × (22/7) × 441
Step 9: Area of sector = (1/6) × 22 × 63
Step 10: Area of sector = (1/6) × 1386 = 231 cm²
Final Answer: 231 cm²
Q7. A circular ring is formed by two concentric circles with radii 5 cm and 3 cm. Find the area of the ring.
Step 1: Area of ring = Area of outer circle − Area of inner circle
Step 2: Area of ring = πR² − πr²
Step 3: Here, R = 5 cm (outer radius), r = 3 cm (inner radius)
Step 4: Area of ring = π(R² − r²) = (22/7)(5² − 3²)
Step 5: Area of ring = (22/7)(25 − 9)
Step 6: Area of ring = (22/7) × 16
Step 7: Area of ring = 352/7 ≈ 50.29 cm²
Final Answer: 352/7 cm² or approximately 50.29 cm²
Q8. A sector of a circle of radius 10 cm contains a central angle of 54°. Find the arc length and area of the sector. (Use π = 3.14)
Step 1: Arc length = (θ/360°) × 2πr
Step 2: Arc length = (54°/360°) × 2 × 3.14 × 10
Step 3: Arc length = (54/360) × 62.8
Step 4: Arc length = 0.15 × 62.8 = 9.42 cm
Step 5: Area of sector = (θ/360°) × πr²
Step 6: Area of sector = (54°/360°) × 3.14 × 10²
Step 7: Area of sector = (54/360) × 3.14 × 100
Step 8: Area of sector = 0.15 × 314 = 47.1 cm²
Final Answer: Arc length = 9.42 cm, Area = 47.1 cm²
Q9. Find the area enclosed between two concentric circles of radii 7 cm and 5 cm. Also find the perimeter of the outer circle. (Use π = 22/7)
Step 1: Area between circles = π(R² − r²)
Step 2: Here, R = 7 cm, r = 5 cm
Step 3: Area = (22/7)(7² − 5²)
Step 4: Area = (22/7)(49 − 25)
Step 5: Area = (22/7) × 24
Step 6: Area = 22 × 24/7 = 528/7 ≈ 75.43 cm²
Step 7: Perimeter of outer circle = 2πR = 2 × (22/7) × 7
Step 8: Perimeter = 2 × 22 = 44 cm
Final Answer: Area between circles = 528/7 cm² ≈ 75.43 cm², Perimeter of outer circle = 44 cm
Areas Related to Circles — Practice MCQs with Answers
Attempt these 19 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. The radius of a circle is 7 cm. What is its area?
- A. 22 cm²
- B. 44 cm²
- C. 154 cm²
- D. 308 cm²
Answer: C. 154 cm² — The area of a circle is calculated using the formula A = πr², where r is the radius.
MCQ 2. What is the circumference of a circle whose diameter is 14 cm?
- A. 22 cm
- B. 44 cm
- C. 88 cm
- D. 154 cm
Answer: B. 44 cm — The circumference of a circle is found using the formula C = πd or C = 2πr, where d is the diameter.
MCQ 3. If the area of a circle is 616 cm², then what is its radius?
- A. 7 cm
- B. 14 cm
- C. 21 cm
- D. 28 cm
Answer: B. 14 cm — Given the area, we can find the radius by rearranging the area formula A = πr².
MCQ 4. A sector of a circle has a radius of 10 cm and a central angle of 60°. What is the area of the sector? (Use π = 3.14)
- A. 52.33 cm²
- B. 78.5 cm²
- C. 104.67 cm²
- D. 314 cm²
Answer: A. 52.33 cm² — The area of a sector is a fraction of the total circle's area, determined by the central angle (θ/360° * πr²).
MCQ 5. The perimeter of a sector of a circle with radius 5.7 cm and central angle 90° is:
- A. 11.4 cm
- B. 20.2 cm
- C. 28.5 cm
- D. 36.1 cm
Answer: B. 20.2 cm — The perimeter of a sector includes the length of the arc and two radii.
MCQ 6. A wheel of diameter 84 cm makes 100 revolutions. What is the distance covered by the wheel? (Use π = 22/7)
- A. 264 m
- B. 528 m
- C. 792 m
- D. 1056 m
Answer: A. 264 m — The distance covered in one revolution is equal to the circumference of the wheel. The total distance is the circumference multiplied by the number of revolutions.
MCQ 7. The minute hand of a clock is 14 cm long. How much area does it sweep in 5 minutes? (Use π = 22/7)
- A. 11.5 cm²
- B. 25.67 cm²
- C. 51.33 cm²
- D. 102.67 cm²
Answer: C. 51.33 cm² — The minute hand completes 360° in 60 minutes, so it sweeps 6° per minute. The area swept is the area of a sector.
MCQ 8. The radii of two circles are 8 cm and 6 cm respectively. What is the radius of the circle having an area equal to the sum of the areas of the two circles?
- A. 10 cm
- B. 14 cm
- C. 24 cm
- D. 100 cm
Answer: A. 10 cm — The area of the new circle will be the sum of the areas of the given circles. Then, use the area formula to find its radius.
MCQ 9. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. What is the length of the arc? (Use π = 22/7)
- A. 11 cm
- B. 22 cm
- C. 33 cm
- D. 44 cm
Answer: B. 22 cm — The length of an arc is a fraction of the total circumference, determined by the central angle.
MCQ 10. A chord of a circle of radius 14 cm subtends a right angle at the centre. What is the area of the minor segment? (Use π = 22/7)
- A. 28 cm²
- B. 56 cm²
- C. 112 cm²
- D. 154 cm²
Answer: B. 56 cm² — The area of a minor segment is the area of the sector minus the area of the triangle formed by the chord and the two radii.
MCQ 11. Four equal circles are described about the four corners of a square so that each circle touches two of its adjacent circles. If the side of the square is 14 cm, what is the area of the region enclosed between the circles? (Use π = 22/7)
- A. 21 cm²
- B. 42 cm²
- C. 84 cm²
- D. 154 cm²
Answer: B. 42 cm² — The area of the shaded region is the area of the square minus the area of four quadrants (which sum up to one full circle).
MCQ 12. If the circumference and the area of a circle are numerically equal, then what is the diameter of the circle?
- A. 2 units
- B. 4 units
- C. 6 units
- D. 8 units
Answer: B. 4 units — Set the circumference formula (2πr) equal to the area formula (πr²) and solve for the radius, then find the diameter.
MCQ 13. A cow is tied with a rope of length 14 m at the corner of a rectangular field of dimensions 20 m x 15 m. What is the maximum area the cow can graze? (Use π = 22/7)
- A. 154 m²
- B. 308 m²
- C. 462 m²
- D. 616 m²
Answer: A. 154 m² — The cow can graze in a quarter-circle (quadrant) since it is tied at a corner of a rectangular field.
MCQ 14. If the area of a sector of a circle with radius 6 cm is 18π cm², what is the corresponding arc length?
- A. 3π cm
- B. 6π cm
- C. 9π cm
- D. 12π cm
Answer: B. 6π cm — The area of a sector is (1/2) * L * r, where L is the arc length and r is the radius. This allows directly finding the arc length.
MCQ 15. A race track is in the form of a ring whose inner and outer circumferences are 440 m and 506 m respectively. What is the width of the track? (Use π = 22/7)
- A. 10.5 m
- B. 11.5 m
- C. 12.5 m
- D. 13.5 m
Answer: A. 10.5 m — Find the inner and outer radii from the given circumferences, then subtract to find the width.
MCQ 16. In a circle of radius 10 cm, a chord subtends an angle of 120° at the centre. The area of the minor segment is: (Use π = 3.14, √3 = 1.73)
- A. 61.4 cm²
- B. 78.5 cm²
- C. 122.8 cm²
- D. 261.67 cm²
Answer: A. 61.4 cm² — The area of a minor segment is the area of the sector minus the area of the triangle formed by the chord and the two radii.
MCQ 17. The areas of two circles are in the ratio 4:9. What is the ratio of their circumferences?
- A. 2:3
- B. 4:9
- C. 16:81
- D. 8:27
Answer: A. 2:3 — If areas are in ratio r₁²:r₂², then radii are in ratio r₁:r₂, and circumferences are also in ratio r₁:r₂.
MCQ 18. If the radius of a circle is doubled, by what factor does its area increase?
- A. 2
- B. 3
- C. 4
- D. 8
Answer: C. 4 — The area of a circle is proportional to the square of its radius. If the radius is doubled (multiplied by 2), the area increases by a factor of 2 squared (4).
MCQ 19. The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour? (Use π = 22/7)
- A. 3125
- B. 3437.5
- C. 4375
- D. 5250
Answer: D. 5250 — First, calculate the distance covered by the car in 10 minutes. Then, find the circumference of the wheel. The number of revolutions is the total distance divided by the circumference.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.