CBSE Class 10 Mathematics Areas Related to Circles MCQs Set 16

Practice MCQs for Class 10 Mathematics Chapter 11 Areas Related to Circles

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Multiple Choice Questions

Question 1: A chord of a circle of radius 14 cm makes an angle of 60° at the centre. The area of the minor sector (use π = 22/7) is:
(a) 102.67 cm²
(b) 205.33 cm²
(c) 51.33 cm²
(d) 308.00 cm²
Show Answer & Explanation

Answer: (a) 102.67 cm²

Explanation:
1. Area = (60/360) x (22/7) x 14² = (1/6)(616) = 102.67 cm².

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 2: For the same circle (radius 14 cm, chord subtending 60° at the centre), the area of the minor segment, taking π = 22/7 and √3 = 1.732, is about:
(a) 84.87 cm²
(b) 25.64 cm²
(c) 12.45 cm²
(d) 17.80 cm²
Show Answer & Explanation

Answer: (d) 17.80 cm²

Explanation:
1. Triangle with two sides 14 and included angle 60° is equilateral: area = (√3/4)(196) = 84.87 cm². Segment = 102.67 - 84.87 = 17.80 cm².

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 3: For the same circle (radius 14 cm, chord subtending 60° at the centre), the area of the major sector (use π = 22/7) is:
(a) 102.67 cm²
(b) 598.20 cm²
(c) 513.33 cm²
(d) 462.00 cm²
Show Answer & Explanation

Answer: (c) 513.33 cm²

Explanation:
1. Circle = (22/7)(196) = 616 cm². Major sector = 616 - 102.67 = 513.33 cm².

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 4: For the same circle (radius 14 cm, chord subtending 60° at the centre), the area of the major segment (use π = 22/7 and √3 = 1.732) is about:
(a) 513.33 cm²
(b) 598.20 cm²
(c) 533.13 cm²
(d) 84.87 cm²
Show Answer & Explanation

Answer: (b) 598.20 cm²

Explanation:
1. Major segment = circle - minor segment = 616 - 17.80 = 598.20 cm².

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 5: ABCP is a quadrant of a circle of radius 14 cm with centre B, and a semicircle is drawn on AC as diameter on the side of P (away from B). The area between the semicircle and the arc APC (use π = 22/7) is:
(a) 56 cm²
(b) 98 cm²
(c) 154 cm²
(d) 196 cm²
Show Answer & Explanation

Answer: (b) 98 cm²

Explanation:
1. AC = 14√2, so the semicircle has radius 7√2 and area (1/2)π(98) = 154 cm². Quadrant = (1/4)π(196) = 154 cm²; triangle ABC = 98 cm². Segment cut by AC = 154 - 98 = 56 cm². The required area = semicircle - segment = 154 - 56 = 98 cm².

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 6: The diameter of a cycle wheel is 28 cm. The number of revolutions it makes in moving 13.2 km (use π = 22/7) is:
(a) 15000
(b) 7500
(c) 12000
(d) 30000
Show Answer & Explanation

Answer: (a) 15000

Explanation:
1. Circumference = π d = (22/7)(28) = 88 cm. Distance = 13.2 km = 1,320,000 cm. Revolutions = 1,320,000 / 88 = 15000.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 7: Two circles touch each other externally. If their radii are 5 cm and 3 cm, the distance between their centres is:
(a) 2 cm
(b) 15 cm
(c) 4 cm
(d) 8 cm
Show Answer & Explanation

Answer: (d) 8 cm

Explanation:
1. For external contact the distance between centres equals the sum of the radii: 5 + 3 = 8 cm.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 8: Two circles touch each other internally. If their radii are 9 cm and 4 cm, the distance between their centres is:
(a) 13 cm
(b) 36 cm
(c) 5 cm
(d) 6.5 cm
Show Answer & Explanation

Answer: (c) 5 cm

Explanation:
1. For internal contact the distance equals the difference of the radii: 9 - 4 = 5 cm.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 9: The angle described by the minute hand of a clock in 1 minute, and by the hour hand in 1 hour, are respectively:
(a) \( 30^\circ \) and \( 6^\circ \)
(b) \( 6^\circ \) and \( 0.5^\circ \)
(c) \( 6^\circ \) and \( 30^\circ \)
(d) \( 12^\circ \) and \( 30^\circ \)
Show Answer & Explanation

Answer: (c) \( 6^\circ \) and \( 30^\circ \)

Explanation:
1. The minute hand turns 360° in 60 minutes = 6° per minute. The hour hand turns 360° in 12 hours = 30° per hour.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 10: A rectangle of sides 5 cm and 15 cm is divided into three squares of equal area. The side of each square is:
(a) 4 cm
(b) 5 cm
(c) 6 cm
(d) 7 cm
Show Answer & Explanation

Answer: (b) 5 cm

Explanation:
1. Area = 75 cm², so each square has area 25 cm² and side 5 cm (the three squares side by side).

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 11: The ratio of the circumference of a circle to its diameter is:
(a) \( \pi \)
(b) \( 2\pi \)
(c) \( \frac{\pi}{2} \)
(d) \( \pi r \)
Show Answer & Explanation

Answer: (a) \( \pi \)

Explanation:
1. Circumference = 2πr and diameter = 2r, so the ratio is π whatever the radius.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 12: The area of a circle inscribed in an equilateral triangle is 154 cm² (use π = 22/7 and √3 = 1.732). The perimeter of the triangle is about:
(a) 84.00 cm
(b) 63.50 cm
(c) 126.00 cm
(d) 72.75 cm
Show Answer & Explanation

Answer: (d) 72.75 cm

Explanation:
1. π r² = 154 gives r = 7 cm. For an equilateral triangle the inradius is side/(2√3), so side = 14√3 and perimeter = 42√3 = 72.74 cm.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 13: The radii of two circles are 8 cm and 6 cm. The radius of the circle whose area equals the sum of their areas is:
(a) 14 cm
(b) 7 cm
(c) 12 cm
(d) 10 cm
Show Answer & Explanation

Answer: (d) 10 cm

Explanation:
1. π R² = π(64 + 36) = 100π, so R = 10 cm.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 14: Each side of an equilateral triangle is 10 cm. With each vertex as centre, a circle of radius 5 cm (half the side) is drawn. The area of the triangle not included in the circles (use π = 3.14 and √3 = 1.732) is about:
(a) 39.25 cm²
(b) 8.05 cm²
(c) 4.05 cm²
(d) 13.75 cm²
Show Answer & Explanation

Answer: (c) 4.05 cm²

Explanation:
1. Triangle = (√3/4)(100) = 43.30 cm². Each circle covers a 60° sector of the triangle; three of them give half a circle = (1/2)(3.14)(25) = 39.25 cm². Remaining = 43.30 - 39.25 = 4.05 cm².

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 15: Semicircles are drawn outward on the legs AB = 3 cm and AC = 4 cm of a right triangle, and a semicircle on the hypotenuse BC is drawn on the side containing A. The total area inside the leg semicircles but outside the hypotenuse semicircle is:
(a) 3 cm²
(b) 6 cm²
(c) 12 cm²
(d) 8 cm²
Show Answer & Explanation

Answer: (b) 6 cm²

Explanation:
1. BC = 5. Area of the leg semicircles + triangle - hypotenuse semicircle = (π/2)(1.5² + 2²) + 6 - (π/2)(2.5²) = (π/2)(6.25) + 6 - (π/2)(6.25) = 6 cm², which equals the area of the triangle.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 16: A semicircular region has radius 14 cm. Its perimeter (use π = 22/7) is:
(a) 72 cm
(b) 44 cm
(c) 58 cm
(d) 100 cm
Show Answer & Explanation

Answer: (a) 72 cm

Explanation:
1. Perimeter = arc + diameter = π r + 2r = 44 + 28 = 72 cm.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 17: The area of a semicircle of radius 14 cm (use π = 22/7) is:
(a) 308 cm²
(b) 616 cm²
(c) 154 cm²
(d) 462 cm²
Show Answer & Explanation

Answer: (a) 308 cm²

Explanation:
1. Area = (1/2)π r² = (1/2)(22/7)(196) = 308 cm².

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 18: The minute hand of a clock is 14 cm long. The area swept by it in 10 minutes (use π = 22/7) is:
(a) 205.33 cm²
(b) 51.33 cm²
(c) 616.00 cm²
(d) 102.67 cm²
Show Answer & Explanation

Answer: (d) 102.67 cm²

Explanation:
1. In 10 minutes it turns 60°. Area = (60/360)(22/7)(14²) = 102.67 cm².

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 19: The length of the minute hand of a clock is 7 cm. The distance its tip travels in 15 minutes (use π = 22/7) is:
(a) 22 cm
(b) 44 cm
(c) 11 cm
(d) 6 cm
Show Answer & Explanation

Answer: (c) 11 cm

Explanation:
1. In 15 minutes the hand turns 90°. Arc = (90/360) x 2 x (22/7) x 7 = 11 cm.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 20: A sector of a circle of radius 21 cm has an angle of 120°. Its arc length (use π = 22/7) is:
(a) 22 cm
(b) 44 cm
(c) 66 cm
(d) 132 cm
Show Answer & Explanation

Answer: (b) 44 cm

Explanation:
1. Arc = (120/360)(2)(22/7)(21) = 44 cm.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 21: A sector of a circle of radius 21 cm has an angle of 120°. Its area (use π = 22/7) is:
(a) 231 cm²
(b) 462 cm²
(c) 693 cm²
(d) 1386 cm²
Show Answer & Explanation

Answer: (b) 462 cm²

Explanation:
1. Area = (120/360)(22/7)(441) = 462 cm².

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

Question 22: The perimeter of a sector of radius 10 cm and angle 90° (use π = 3.14) is:
(a) 35.7 cm
(b) 25.7 cm
(c) 31.4 cm
(d) 45.7 cm
Show Answer & Explanation

Answer: (a) 35.7 cm

Explanation:
1. Arc = (1/4)(2)(3.14)(10) = 15.7 cm. Perimeter = 15.7 + 10 + 10 = 35.7 cm.

Teacher's Note:
1. Circle: circumference = 2π r, area = π r². Sector of angle θ: arc = (θ/360) x 2π r, area = (θ/360) x π r².
2. Segment area = sector area - triangle area (for a minor segment).

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Practice MCQs for Class 10 Mathematics Chapter 11 Areas Related to Circles

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