Maths · Mensuration · chapter 11 of 14

Class 10 Maths Chapter 11: Areas Related to Circles worksheet with answer key

A 20-mark worksheet on Areas Related to Circles for about 40 minutes of class or homework: 5 multiple-choice, 3 very short answer and 3 short answer questions, written from the NCERT chapter and checked by a second reviewer. Print the PDF for the class; the answer key carries the marking scheme. Teachers may photocopy it freely.

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Question 1
Two radii of a circle make an angle of 120° at the centre. The angle of the major sector they form is
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Answer: (b) 240°

The major sector's angle is 360° − 120° = 240°. 60° is 180° − 120°, as if the angles made a straight line.

Question 2
OA and OB are radii of a circle of radius 12 cm with ∠AOB = 90°. The area of triangle OAB is
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Answer: (a) 72 cm²

The triangle is right-angled at O with legs 12 cm and 12 cm: ½ × 12 × 12 = 72 cm². 144 forgets the ½; 36√3 is the 60° (equilateral) formula.

Question 3
OA and OB are radii of a circle of radius 10 cm with ∠AOB = 60°. The area of triangle OAB is
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Answer: (d) 25√3 cm²

With a 60° angle between equal sides the triangle is equilateral of side 10 cm: (√3/4) × 10² = 25√3 cm². 50 cm² treats it as right-angled.

Question 4
Two radii of a circle of radius 28 cm are perpendicular. The area of the major sector they form is. Take π = 22/7.
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Answer: (d) 1848 cm²

Circle = (22/7) × 28² = 2464 cm²; minor sector = 2464/4 = 616 cm²; major sector = 2464 − 616 = 1848 cm².

Question 5
The angle of a sector is kept the same and its radius is doubled. Then
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Answer: (b) its arc length doubles and its area becomes four times

Arc = (θ/360) × 2πr grows as r, so it doubles; area = (θ/360) × πr² grows as r², so it becomes 2² = 4 times.

Question 6 · 2 marks
Two radii at 60° to each other divide a circular rangoli of radius 42 cm into a minor and a major sector. Find the area of the major sector. Take π = 22/7.
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Circle = (22/7) × 42² = 5544 cm²; minor sector = 5544/6 = 924 cm²; major sector = 5544 − 924 = 4620 cm² (or (300/360) × 5544).

  • minor sector 924 cm², or the major angle 300° (1)
  • major sector 4620 cm² (1)
Question 7 · 2 marks
A round roti with centre O and radius 21 cm is cut in two along a straight line AB, where ∠AOB = 90°. Find the area of the smaller piece. Take π = 22/7.
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The smaller piece is the minor segment. Sector = (90/360) × (22/7) × 441 = 346.5 cm². Triangle OAB = ½ × 21 × 21 = 220.5 cm². Segment = 346.5 − 220.5 = 126 cm².

  • sector 346.5 cm² and triangle 220.5 cm² (1)
  • smaller piece 126 cm² (1)
Question 8 · 2 marks
A wedge of cheese has a top face in the shape of a sector of radius 28 mm and angle 90°. Find the length of the curved edge and the area of the top face. Take π = 22/7.
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Arc = (90/360) × 2 × (22/7) × 28 = 44 mm. Area = (90/360) × (22/7) × 28² = 616 mm².

  • arc length 44 mm (1)
  • area 616 mm² (1)
Question 9 · 3 marks
A chord of a circle of radius 12 cm is equal to the radius. Find the area of the minor segment it cuts off. (Take π = 3.14 and √3 = 1.73.)
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A chord equal to the radius makes an equilateral triangle with the two radii, so it subtends 60°. Sector = (60/360) × 3.14 × 144 = 75.36 cm². Triangle = (√3/4) × 144 = 36 × 1.73 = 62.28 cm². Segment = 75.36 − 62.28 = 13.08 cm².

  • angle 60° from the equilateral triangle (1)
  • sector 75.36 cm² and triangle 62.28 cm² (1)
  • segment 13.08 cm² (1)
Question 10 · 3 marks
A lawn sprinkler stands at an outer corner of a building, so the building blocks a 90° part of its turn and it sprays through the rest of the full circle. Its spray reaches 14 m. Find the area watered and the length of the outer edge of the watered region. Take π = 22/7.
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The sprinkler turns through 360° − 90° = 270°, a major sector. Area = (270/360) × (22/7) × 14² = ¾ × 616 = 462 m². Edge = (270/360) × 2 × (22/7) × 14 = ¾ × 88 = 66 m.

  • angle 270° (1)
  • area 462 m² (1)
  • arc 66 m (1)
Question 11 · 3 marks
A circular metal plate of radius 14 cm has two circular holes, each of radius 3.5 cm, punched in it, as in the figure. Find the area of metal left. Take π = 22/7.
O14 cm3.5 cm
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Plate = (22/7) × 14² = 616 cm². One hole = (22/7) × 3.5² = 38.5 cm²; two holes = 77 cm². Metal left = 616 − 77 = 539 cm².

  • plate 616 cm² (1)
  • two holes 77 cm² (1)
  • metal left 539 cm² (1)
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