RowQ
The Vault
RowQ
The Vault
CBSE Class 10 Maths · 11 questions · 26 marks
This chapter is short but it rewards neat bookkeeping more than clever ideas. Nearly everything follows from two formulas, area πr² and circumference 2πr, scaled down by the fraction θ/360 when you only want a slice of the circle. The marks are usually lost by mixing up radius with diameter, or by forgetting that a segment is a sector with a triangle cut away, so write down r and θ clearly before you start computing.
The area of a sector of a circle of radius 21 cm with a central angle of 60° is (take π = 22/7):
Answer
The area is 231 cm². Area of sector = (θ/360) × πr². = (60/360) × (22/7) × 21 × 21 = (1/6) × (22/7) × 441 = (1/6) × 22 × 63 = (1/6) × 1386 = 231 cm².
The circumference of a circle is 88 cm. Its area is (take π = 22/7):
Answer
The area is 616 cm². First find the radius from the circumference: 2πr = 88. 2 × (22/7) × r = 88, so (44/7)r = 88. r = 88 × 7/44 = 14 cm. Now area = πr² = (22/7) × 14 × 14 = 22 × 2 × 14 = 616 cm².
In a circle of radius 14 cm, the length of an arc that subtends a right angle at the centre is (take π = 22/7):
Answer
The arc length is 22 cm. Arc length = (θ/360) × 2πr with θ = 90°. = (90/360) × 2 × (22/7) × 14 = (1/4) × (44/7) × 14 = (1/4) × 88 = 22 cm. The value 154 cm² would be the area of that quadrant, not the arc length.
A cart wheel of diameter 70 cm rolls without slipping. The distance it covers in 100 complete revolutions is (take π = 22/7):
Answer
The distance is 220 m. In one revolution the wheel covers a distance equal to its circumference. Circumference = πd = (22/7) × 70 = 220 cm. In 100 revolutions, distance = 100 × 220 = 22000 cm. Converting, 22000 cm = 22000/100 m = 220 m.
Assertion (A): If the radius of a circle is doubled, its area becomes four times the original area. Reason (R): The area of a circle is directly proportional to the square of its radius.
Answer
Both A and R are true and R is the correct explanation of A. Reason R is correct because area = πr², and π is a constant, so area ∝ r². Now test Assertion A using this relation. Let the original radius be r, so the original area is πr². With the radius doubled, the new area = π(2r)² = π × 4r² = 4πr². This is four times the original area, so A is true. The squaring in R is precisely what turns a factor of 2 in the radius into a factor of 4 in the area, so R explains A.
The circumference of a circle is 44 cm. Find the area of a quadrant of this circle. (Take π = 22/7)
Answer
From 2πr = 44: 2 × (22/7) × r = 44, so (44/7)r = 44 and r = 7 cm. A quadrant is one fourth of the circle, so its area = (1/4)πr². = (1/4) × (22/7) × 49 = (1/4) × 154 = 38.5 cm².
A sector of a circle of radius 10.5 cm has a central angle of 120°. Find (i) its area and (ii) its perimeter. (Take π = 22/7)
Answer
(i) Area of sector = (θ/360) × πr² = (120/360) × (22/7) × 10.5 × 10.5 = (1/3) × (22/7) × 110.25 = (1/3) × 346.5 = 115.5 cm². (ii) Arc length = (120/360) × 2πr = (1/3) × 2 × (22/7) × 10.5 = (1/3) × 66 = 22 cm. Perimeter of the sector = arc + the two bounding radii = 22 + 10.5 + 10.5 = 43 cm.
A flat washer is shaped like a ring between two concentric circles of radii 12 cm and 5 cm. Find the area of the metal in the washer. (Take π = 3.14)
Answer
Area of ring = πR² - πr² = π(R² - r²), with R = 12 cm and r = 5 cm. R² - r² = 144 - 25 = 119. Area = 3.14 × 119 = 373.66 cm². So the washer contains 373.66 cm² of metal.
A square metal plate ABCD has side 28 cm. A quadrant of radius 7 cm is cut away at each of the four corners, and in addition a circular hole of diameter 14 cm is punched out at the centre of the plate. Find the area of the metal remaining. (Take π = 22/7)
Answer
Step 1 — Area of the square plate = side² = 28 × 28 = 784 cm². Step 2 — Area removed by the four corner quadrants. Each quadrant has radius 7 cm and is one fourth of a circle, so four of them together make one full circle of radius 7 cm. Combined area = πr² = (22/7) × 7 × 7 = 154 cm². Step 3 — Area of the central circular hole. Its diameter is 14 cm, so its radius is 7 cm. Area = (22/7) × 7 × 7 = 154 cm². Step 4 — Total area removed = 154 + 154 = 308 cm². Step 5 — Remaining metal = 784 - 308 = 476 cm². So 476 cm² of metal is left in the plate.
A chord of a circle of radius 14 cm subtends a right angle at the centre. Find (i) the area of the corresponding minor sector, (ii) the area of the minor segment, and (iii) the area of the major segment. (Take π = 22/7)
Answer
Let the centre be O and the chord be AB with ∠AOB = 90°, and OA = OB = 14 cm. (i) Area of minor sector OAB = (θ/360) × πr² = (90/360) × (22/7) × 14 × 14 = (1/4) × (22/7) × 196 = (1/4) × 616 = 154 cm². (ii) ΔOAB is right-angled at O, with the two radii as its perpendicular sides. Area of ΔOAB = (1/2) × OA × OB = (1/2) × 14 × 14 = 98 cm². Area of minor segment = area of sector - area of triangle = 154 - 98 = 56 cm². (iii) Area of the whole circle = πr² = (22/7) × 196 = 616 cm². Area of major segment = area of circle - area of minor segment = 616 - 56 = 560 cm². Check: 56 + 560 = 616 cm², which matches the full circle. ✔
Read the following and answer the questions that follow: A wall clock in a school library has a minute hand of length 14 cm, measured from the centre pivot to the tip. As time passes, the hand sweeps out sectors of a circle of radius 14 cm. (Take π = 22/7) (a) Find the area swept by the minute hand in 15 minutes. (b) Find the area swept by the minute hand in 20 minutes. (c) Find the distance travelled by the tip of the hand in 30 minutes. (d) Find the angle turned by the hand in 25 minutes and the area it sweeps in that time.
Answer
The minute hand turns through 360° in 60 minutes, so it turns 6° per minute. (a) In 15 minutes the hand turns 15 × 6° = 90°. Area = (90/360) × (22/7) × 14 × 14 = (1/4) × 616 = 154 cm². (b) In 20 minutes the hand turns 20 × 6° = 120°. Area = (120/360) × 616 = (1/3) × 616 = 205.33 cm² (correct to two decimal places). (c) In 30 minutes the hand turns 180°, so the tip travels along half the circumference. Circumference = 2πr = 2 × (22/7) × 14 = 88 cm. Distance travelled = 88/2 = 44 cm. (d) In 25 minutes the hand turns 25 × 6° = 150°. Area = (150/360) × 616 = (5/12) × 616 = 3080/12 = 256.67 cm² (correct to two decimal places).
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