CBSE Class 10 Mathematics Surface Areas and Volumes MCQs Set 27

Mathematics Objective Questions and Answers: Chapter 12 Surface Areas and Volumes

Review structured MCQ sets for Class 10 Mathematics Chapter 12 Surface Areas and Volumes. Built according to official CBSE guidelines, these downloadable questions support daily revision and core concept reinforcement.

Download Chapter 12 Surface Areas and Volumes MCQs with Answers

View or download the dedicated Chapter 12 Surface Areas and Volumes MCQ resource below. Practicing these 50 objective questions regularly builds familiarity with standard exam patterns and helps secure higher marks in final Mathematics evaluations.

Multiple Choice Questions

Question 1: The volume of a cube (in cm³) plus three times the total length of its edges (in cm) is equal to twice its surface area (in cm²). The length of its diagonal is
(a) 6 cm
(b) 6√3 cm
(c) 3√6 cm
(d) 6√6 cm
Show Answer & Explanation

Answer: (b) 6√3 cm

Explanation:
1. Let the edge be s. Volume + 3 × (12s) = 2 × (6s²), so s³ + 36s = 12s².
2. Dividing by s: s² − 12s + 36 = 0, so (s − 6)² = 0 and s = 6.
3. Diagonal = s√3 = 6√3 cm.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 2: An aquarium has a rectangular base measuring 100 cm by 40 cm and height 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base 40 cm by 20 cm and height 10 cm is placed in the aquarium (fully under water). The water level rises by
(a) 0.5 cm
(b) 1 cm
(c) 1.5 cm
(d) 2 cm
Show Answer & Explanation

Answer: (d) 2 cm

Explanation:
1. Volume of the brick = 40 × 20 × 10 = 8000 cm³.
2. Rise in level = volume ÷ base area of the aquarium = 8000 ÷ (100 × 40) = 2 cm.

Teacher's Note:
1. The displaced water volume equals the volume of the submerged brick.
2. The level rises over the whole base of the aquarium.

Question 3: The curved surface area of a cylinder is 264 m² and its volume is 924 m³. The ratio of its diameter to its height is
(a) 7 : 6
(b) 6 : 7
(c) 3 : 7
(d) 7 : 3
Show Answer & Explanation

Answer: (d) 7 : 3

Explanation:
1. V/CSA = (πr²h)/(2πrh) = r/2, so r = 2 × 924/264 = 7 m.
2. CSA = 2πrh = 264 gives h = 264/(2π × 7) = 6 m.
3. Diameter : height = 14 : 6 = 7 : 3.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 4: A cylindrical tank has base radius 7 m and height 9 m. If 2/3 of the tank is filled with water, the volume of water is (π = 22/7)
(a) 1386 m³
(b) 462 m³
(c) 824 m³
(d) 924 m³
Show Answer & Explanation

Answer: (d) 924 m³

Explanation:
1. Volume of the tank = (22/7) × 49 × 9 = 1386 m³.
2. Water = 2/3 × 1386 = 924 m³.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 5: Two steel sheets, each of length a₁ and breadth a₂, are rolled to form two right circular cylinders: one with volume V₁ and height a₂, the other with volume V₂ and height a₁. Then
(a) V₁ = V₂
(b) a₁V₁ = a₂V₂
(c) a₂V₁ = a₁V₂
(d) V₁²/a₁ = V₂²/a₂
Show Answer & Explanation

Answer: (c) a₂V₁ = a₁V₂

Explanation:
1. First cylinder: height a₂, circumference a₁, so r₁ = a₁/(2π) and V₁ = π r₁² a₂ = a₁²a₂/(4π).
2. Second: height a₁, circumference a₂, so V₂ = a₂²a₁/(4π).
3. V₁/V₂ = a₁/a₂, that is a₂V₁ = a₁V₂.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 6: The slant height of a cone is increased by P% while the radius stays the same. The curved surface area is increased by
(a) P%
(b) P²%
(c) 2P%
(d) none of these
Show Answer & Explanation

Answer: (a) P%

Explanation:
1. CSA = πrl is proportional to l when r is fixed.
2. So the curved surface area also increases by P%.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 7: It is required to construct a conical circus tent of radius 21 m and slant height 35 m. The width of the canvas is 3 m. The length of the canvas required is
(a) 700 m
(b) 1250 m
(c) 776.5 m
(d) 770 m
Show Answer & Explanation

Answer: (d) 770 m

Explanation:
1. Curved surface area = πrl = (22/7) × 21 × 35 = 2310 m².
2. Length of canvas = area ÷ width = 2310/3 = 770 m.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 8: A hemispherical dome has an inner diameter of 28 m. Its volume (in m³) is (π = 22/7)
(a) 6186.60
(b) 5749.33
(c) 7099.33
(d) 7459.33
Show Answer & Explanation

Answer: (b) 5749.33

Explanation:
1. r = 14 m. Volume = (2/3)πr³ = (2/3) × (22/7) × 2744.
2. = 5749.33 m³.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 9: The number of cubes of volume 4 cubic units that can be cut from a cube with a surface area of 96 square units is
(a) 4
(b) 8
(c) 12
(d) 16
Show Answer & Explanation

Answer: (d) 16

Explanation:
1. 6a² = 96 gives a = 4, so the volume of the big cube = 64.
2. Number of small cubes = 64/4 = 16.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 10: A lead ball of radius 24 cm is melted down and recast into smaller balls of radius 6 cm. The number of complete smaller balls is
(a) 4
(b) 16
(c) 36
(d) 64
Show Answer & Explanation

Answer: (d) 64

Explanation:
1. Number = (24/6)³ = 4³ = 64.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 11: A sphere of radius 3 cm is dropped into a cylindrical vessel of radius 4 cm and is completely submerged. The height to which the water rises is
(a) 2.35 cm
(b) 2.30 cm
(c) 2.25 cm
(d) 2.15 cm
Show Answer & Explanation

Answer: (c) 2.25 cm

Explanation:
1. Volume of the sphere = (4/3)π(27) = 36π cm³.
2. Rise × π × 4² = 36π, so rise = 36/16 = 2.25 cm.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 12: Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter 2 cm and height 16 cm. The diameter of each sphere is
(a) 4 cm
(b) 3 cm
(c) 2 cm
(d) 6 cm
Show Answer & Explanation

Answer: (c) 2 cm

Explanation:
1. Volume of the cylinder = π × 1² × 16 = 16π.
2. 12 × (4/3)πr³ = 16π gives 16r³ = 16, so r = 1 cm.
3. Diameter = 2 cm.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 13: A sphere of radius 6 cm is dropped into a cylindrical vessel partly filled with water. The radius of the vessel is 8 cm. If the sphere is submerged completely, the surface of the water rises by
(a) 4.5 cm
(b) 4 cm
(c) 3 cm
(d) 2 cm
Show Answer & Explanation

Answer: (a) 4.5 cm

Explanation:
1. Volume of the sphere = (4/3)π × 216 = 288π.
2. Rise = 288π/(π × 64) = 4.5 cm.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 14: A metallic spherical shell of internal and external diameters 8 cm and 12 cm is melted and recast into a cone of base diameter 8 cm. The height of the cone is
(a) 114 cm
(b) 76 cm
(c) 38 cm
(d) 19 cm
Show Answer & Explanation

Answer: (c) 38 cm

Explanation:
1. Volume of the shell = (4/3)π(6³ − 4³) = (4/3)π × 152.
2. Cone: (1/3)π × 4² × h = (4/3)π × 152, so 16h = 608 and h = 38 cm.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 15: A solid piece of iron of dimensions 66 cm × 49 cm × 12 cm is moulded into a sphere. The radius of the sphere is (π = 22/7)
(a) 7 cm
(b) 3 cm
(c) 21 cm
(d) 14 cm
Show Answer & Explanation

Answer: (c) 21 cm

Explanation:
1. Volume = 66 × 49 × 12 = 38808 cm³.
2. (4/3)(22/7)r³ = 38808, so r³ = 38808 × 21/88 = 9261 and r = 21 cm.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 16: The number of solid spheres, each of diameter 6 cm, that can be moulded to form a solid metal cylinder of height 45 cm and diameter 4 cm is
(a) 3
(b) 4
(c) 5
(d) 6
Show Answer & Explanation

Answer: (c) 5

Explanation:
1. Volume of the cylinder = π × 4 × 45 = 180π.
2. Volume of a sphere = (4/3)π × 27 = 36π.
3. Number = 180π/36π = 5.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 17: A cylinder circumscribes a hemisphere such that their bases are common. The ratio of the volume of the hemisphere to that of the cylinder is
(a) 1 : 3
(b) 1 : 2
(c) 2 : 3
(d) 3 : 4
Show Answer & Explanation

Answer: (c) 2 : 3

Explanation:
1. Hemisphere: (2/3)πr³. The cylinder has height r: πr³.
2. Ratio = 2 : 3.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 18: A medicine capsule is in the shape of a cylinder of diameter 0.5 cm with a hemisphere stuck to each end. The length of the entire capsule is 2 cm. The capacity of the capsule is (π = 22/7)
(a) 0.36 cm³
(b) 0.40 cm³
(c) 0.45 cm³
(d) 0.30 cm³
Show Answer & Explanation

Answer: (a) 0.36 cm³

Explanation:
1. r = 0.25 cm, so the cylindrical part has length 2 − 0.5 = 1.5 cm.
2. Volume = π(0.25)²(1.5) + (4/3)π(0.25)³ = π(0.09375 + 0.020833) = 0.114583π.
3. With π = 22/7, volume ≈ 0.36 cm³.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 19: A solid consists of a circular cylinder with an exactly fitting right circular cone placed at the top. If the height of the cone is h and the total volume of the solid is 3 times the volume of the cone, then the height of the cylinder is
(a) 2h
(b) 2h/3
(c) 3h/2
(d) 4h
Show Answer & Explanation

Answer: (b) 2h/3

Explanation:
1. Let the common radius be r and cylinder height H. πr²H + (1/3)πr²h = 3 × (1/3)πr²h = πr²h.
2. H = h − h/3 = 2h/3.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 20: A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into 12 toys, each a right circular cone mounted on a hemisphere. The height of the conical part is 3 times its radius. The total height of a toy is
(a) 9 cm
(b) 12 cm
(c) 15 cm
(d) 18 cm
Show Answer & Explanation

Answer: (b) 12 cm

Explanation:
1. Volume of the cylinder = π × 36 × 15 = 540π, so each toy has volume 45π.
2. Toy: cone height 3r and hemisphere radius r: (1/3)πr²(3r) + (2/3)πr³ = (5/3)πr³ = 45π, so r³ = 27 and r = 3 cm.
3. Total height = 3r + r = 4r = 12 cm.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 21: A tent is a cylinder of diameter 4.2 m and height 4 m surmounted by a cone of equal base and height 2.8 m. The cost of canvas at Rs. 100 per sq. m is (π = 22/7)
(a) Rs. 7590
(b) Rs. 7950
(c) Rs. 7580
(d) Rs. 7850
Show Answer & Explanation

Answer: (a) Rs. 7590

Explanation:
1. r = 2.1 m; the slant height of the cone l = √(2.1² + 2.8²) = 3.5 m.
2. Canvas = curved surface of the cylinder + curved surface of the cone = 2π(2.1)(4) + π(2.1)(3.5) = π(2.1)(11.5) = (22/7)(24.15) = 75.9 m².
3. Cost = 75.9 × 100 = Rs. 7590.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 22: A cone is divided into two parts by a plane through the midpoint of its axis, parallel to its base. The ratio of the volumes of the two parts is
(a) 1 : 3
(b) 1 : 7
(c) 1 : 8
(d) 1 : 9
Show Answer & Explanation

Answer: (b) 1 : 7

Explanation:
1. The small cone on top is similar to the whole cone with ratio 1 : 2, so its volume is 1/8 of the whole.
2. The frustum has 7/8, so the ratio is 1 : 7.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 23: The radii of the circular ends of a bucket of height 40 cm are 35 cm and 14 cm. The volume of the bucket (in cm³) is (π = 22/7)
(a) 60060
(b) 80080
(c) 70070
(d) 80160
Show Answer & Explanation

Answer: (b) 80080

Explanation:
1. V = (π/3) h (R² + Rr + r²) = (π/3) × 40 × (1225 + 490 + 196) = (π/3) × 40 × 1911.
2. = (22/7) × 40 × 637 = 80080 cm³.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 24: A right circular cone is cut at the middle of its height by a plane parallel to the base. The smaller cone is A and the remaining part is B. Then
(a) Vol. A < Vol. B
(b) Vol. A = Vol. B
(c) Vol. A > Vol. B
(d) Vol. A = ½ (Vol. B)
Show Answer & Explanation

Answer: (a) Vol. A < Vol. B

Explanation:
1. The small cone is similar with ratio 1 : 2, so Vol. A = 1/8 of the whole and Vol. B = 7/8.
2. Hence Vol. A < Vol. B.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

Question 25: The radii of the circular ends of a bucket of height 15 cm are 14 cm and r cm (r < 14). If the volume of the bucket is 5390 cm³, then r is (π = 22/7)
(a) 4 cm
(b) 5 cm
(c) 6 cm
(d) 7 cm
Show Answer & Explanation

Answer: (d) 7 cm

Explanation:
1. (π/3) × 15 × (196 + 14r + r²) = 5390, so 5 × (22/7)(196 + 14r + r²) = 5390.
2. 196 + 14r + r² = 343, so r² + 14r − 147 = 0 and (r + 21)(r − 7) = 0.
3. r = 7 cm.

Teacher's Note:
1. Write down the formula for the solid before substituting any numbers.
2. When a solid is melted and recast, its volume stays the same.

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Practice MCQs for Class 10 Mathematics Chapter 12 Surface Areas and Volumes

About Chapter 12 Surface Areas and Volumes MCQs for Class 10 Mathematics

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