Mensuration — 2D & 3D — Practice

Question 1

A rhombus has diagonals 16 cm and 30 cm. Its perimeter is:
(a) 60 cm  (b) 64 cm  (c) 68 cm  (d) 72 cm

Accepted Answer

Half-diagonals 8 and 15 → side = √(64+225) = 17 (8-15-17 triple).

Perimeter = 4 × 17 = 68 cm.

Answer: (c) 68 cm.

Question 2

If the radius of a circle is increased by 50%, its area increases by:
(a) 50%  (b) 100%  (c) 125%  (d) 150%

Accepted Answer

Factor 1.5² = 2.25 → +125%.

Answer: (c) 125%.

Question 3

The curved surface area of a cone with radius 21 cm and height 28 cm is:
(a) 2,310 cm²  (b) 2,110 cm²  (c) 2,530 cm²  (d) 1,980 cm²

Accepted Answer

l = √(21² + 28²) = √(441+784) = √1225 = 35 (3-4-5 × 7).

CSA = πrl = (22/7) × 21 × 35 = 22 × 105 = 2,310 cm².

Answer: (a).

Question 4

A solid metallic sphere of radius 12 cm is melted into small spheres of radius 3 cm each. The number of small spheres is:
(a) 16  (b) 27  (c) 48  (d) 64

Accepted Answer

n = (R/r)³ = (12/3)³ = 64.

Answer: (d) 64.

Question 5

A cone, a hemisphere and a cylinder stand on the same base and have the same height (equal to the radius). The ratio of their volumes is:
(a) 1:2:3  (b) 2:1:3  (c) 3:2:1  (d) 1:3:2

Accepted Answer

⅓πr³ : ⅔πr³ : πr³ = 1 : 2 : 3.

Answer: (a) 1:2:3.

Question 6

A copper wire bent into a square of side 33 cm is re-bent into a circle. The radius of the circle is:
(a) 21 cm  (b) 22 cm  (c) 28 cm  (d) 42 cm

Accepted Answer

Wire length = 4 × 33 = 132 cm = 2πr.

r = 132 × 7/(2 × 22) = 21 cm.

Answer: (a) 21 cm.

Question 7

A sphere of radius 6 cm is melted and drawn into a wire of radius 0.2 cm. The length of the wire is:
(a) 36 m  (b) 72 m  (c) 12 m  (d) 24 m

Accepted Answer

Volume: (4/3)π(216) = π(0.04)L → 288 = 0.04L → L = 7,200 cm = 72 m.

Answer: (b) 72 m.

Question 8

A bucket is a frustum with top radius 14 cm, bottom radius 7 cm and height 24 cm. Its slant height is:
(a) 24 cm  (b) 25 cm  (c) 26 cm  (d) 30 cm

Accepted Answer

l = √(h² + (R−r)²) = √(576 + 49) = √625 = 25 cm (7-24-25).

Answer: (b) 25 cm.

Question 9

The longest rod that can fit inside a cubical room of side 10 m has length:
(a) 10√2 m  (b) 10√3 m  (c) 20 m  (d) 10 m

Accepted Answer

Space diagonal of a cube = a√3 = 10√3 m.

Answer: (b). (10√2 is the face diagonal trap.)

Question 10

The volume of a sphere is numerically equal to its surface area. Its diameter is:
(a) 3 units  (b) 4 units  (c) 6 units  (d) 8 units

Accepted Answer

(4/3)πr³ = 4πr² → r = 3 → diameter 6.

Answer: (c) 6 units. (Same pattern for the cube gives edge 6.)

Question 11

A hemispherical tank of radius 6 m is full of water, emptied by a pipe at the rate of 12π m³ per hour. Time to empty it:
(a) 8 h  (b) 10 h  (c) 12 h  (d) 16 h

Accepted Answer

V = (2/3)π(216) = 144π m³.

Time = 144π/12π = 12 hours.

Answer: (c) 12 h.

Practice — Mensuration — 2D & 3D

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