Practice Quantitative Aptitude questions on Volume and Surface Area, covering cubes, cuboids, cylinders, cones, spheres and hemispheres, including volume calculations, curved and total surface area, and unit conversions. Takes about 20–25 minutes and is useful for placement tests, TCS NQT, SSC, banking and MBA entrance exams.
Question 1
An open water tank measures 2.4 m × 1.8 m × 1.3 m externally. Its walls are 10 cm thick and it holds 4224 litres. Find the thickness of the bottom of the tank.
Explanation
Capacity = 4224 litres = 4.224 m3.
Internal length = 2.4 − 2 × 0.1 = 2.2 m and internal width = 1.8 − 2 × 0.1 = 1.6 m, so the internal base area = 2.2 × 1.6 = 3.52 m2.
Internal depth = 4.224 ÷ 3.52 = 1.2 m.
Thickness of the bottom = 1.3 − 1.2 = 0.1 m = 10 cm.
Internal length = 2.4 − 2 × 0.1 = 2.2 m and internal width = 1.8 − 2 × 0.1 = 1.6 m, so the internal base area = 2.2 × 1.6 = 3.52 m2.
Internal depth = 4.224 ÷ 3.52 = 1.2 m.
Thickness of the bottom = 1.3 − 1.2 = 0.1 m = 10 cm.
Shortcut Method
Inner base = 2.2 × 1.6 = 3.52 m2
Depth = 4.224 ÷ 3.52 = 1.2 m
∴ Bottom = 1.3 − 1.2 = 0.1 m = 10 cm ✓
Depth = 4.224 ÷ 3.52 = 1.2 m
∴ Bottom = 1.3 − 1.2 = 0.1 m = 10 cm ✓
Question 2
A cylinder has a curved surface area of 528 cm2 and a volume of 2772 cm3. What is the ratio of its radius to its height? (Take π = 227.)
Explanation
Curved surface area = 2π r h = 528 and volume = π r2 h = 2772.
Dividing: r2 = 2772528 = 5.25, so r = 10.5 cm.
Then 2 × 227 × 10.5 × h = 528, so 66h = 528 and h = 8 cm.
Radius : height = 10.5 : 8 = 21 : 16.
Dividing: r2 = 2772528 = 5.25, so r = 10.5 cm.
Then 2 × 227 × 10.5 × h = 528, so 66h = 528 and h = 8 cm.
Radius : height = 10.5 : 8 = 21 : 16.
Shortcut Method
V ÷ CSA = r2 = 2772 ÷ 528 = 5.25 → r = 10.5
CSA = 66h = 528 → h = 8
∴ r : h = 10.5 : 8 = 21 : 16 ✓
CSA = 66h = 528 → h = 8
∴ r : h = 10.5 : 8 = 21 : 16 ✓
Question 3
A solid cube of side 15 cm is cut into 125 identical small cubes. What is the ratio of the total surface area of all the small cubes to the surface area of the original cube?
Explanation
Volume of the original cube = 153 = 3375 cm3, so each small cube has volume 3375 ÷ 125 = 27 cm3 and edge 3 cm.
Total surface area of the small cubes = 125 × 6 × 32 = 6750 cm2.
Surface area of the original cube = 6 × 152 = 1350 cm2.
Ratio = 6750 : 1350 = 5 : 1.
Total surface area of the small cubes = 125 × 6 × 32 = 6750 cm2.
Surface area of the original cube = 6 × 152 = 1350 cm2.
Ratio = 6750 : 1350 = 5 : 1.
Shortcut Method
125 cubes → edge ratio 15 : 3 = 5 : 1
Total area ratio = 125 × 32 : 152 = 1125 : 225
∴ 5 : 1 ✓
Total area ratio = 125 × 32 : 152 = 1125 : 225
∴ 5 : 1 ✓
Question 4
A boundary wall 12 m long, 4.8 m high and 36 cm thick is to be built using bricks measuring 24 cm × 12 cm × 8 cm. How many bricks are required?
Explanation
Volume of the wall = 1200 × 480 × 36 = 2,07,36,000 cm3 (converting metres to centimetres).
Volume of one brick = 24 × 12 × 8 = 2304 cm3.
Number of bricks = 2,07,36,000 ÷ 2304 = 9000.
Volume of one brick = 24 × 12 × 8 = 2304 cm3.
Number of bricks = 2,07,36,000 ÷ 2304 = 9000.
Shortcut Method
Wall = 1200 × 480 × 36 = 20,736,000 cm3
Brick = 24 × 12 × 8 = 2304
∴ 20,736,000 ÷ 2304 = 9000 bricks ✓
Brick = 24 × 12 × 8 = 2304
∴ 20,736,000 ÷ 2304 = 9000 bricks ✓
Question 5
A cylindrical jar of radius 6 cm is partly filled with water. A solid spherical ball of radius 3 cm is dropped into it and sinks completely without any water spilling out. By how many centimetres does the water level rise?
Explanation
Volume of the ball = 43π r3 = 43π × 27 = 36π cm3.
Let the rise in water level be h. Volume of water raised = π × 62 × h = 36π h.
36π h = 36π, so h = 1 cm.
Let the rise in water level be h. Volume of water raised = π × 62 × h = 36π h.
36π h = 36π, so h = 1 cm.
Shortcut Method
Ball = 43π × 27 = 36π
Jar: π × 36 × h = 36π h
∴ h = 1 cm ✓
Jar: π × 36 × h = 36π h
∴ h = 1 cm ✓
Question 6
A cone has a base radius of 3.5 cm and a height of 12 cm. Its total surface area, taking π = 227, is:
Explanation
Slant height l = √r2 + h2 = √12.25 + 144 = √156.25 = 12.5 cm.
Total surface area = π r(l + r) = 227 × 3.5 × (12.5 + 3.5)
= 227 × 3.5 × 16 = 22 × 0.5 × 16 = 176 cm2.
Total surface area = π r(l + r) = 227 × 3.5 × (12.5 + 3.5)
= 227 × 3.5 × 16 = 22 × 0.5 × 16 = 176 cm2.
Shortcut Method
(3.5, 12, 12.5) → l = 12.5
TSA = π r(l + r) = 227 × 3.5 × 16
∴ TSA = 176 cm2 ✓
TSA = π r(l + r) = 227 × 3.5 × 16
∴ TSA = 176 cm2 ✓
Question 7
The radius of a sphere is decreased by 20%. By what percentage does its volume decrease?
Explanation
Volume of a sphere = 43π r3, so volume is proportional to r3.
New radius = 0.8r, so the new volume = (0.8)3 = 0.512 times the old volume.
Decrease = 1 − 0.512 = 0.488 = 48.8%.
New radius = 0.8r, so the new volume = (0.8)3 = 0.512 times the old volume.
Decrease = 1 − 0.512 = 0.488 = 48.8%.
Shortcut Method
V ∝ r3 → factor = 0.83 = 0.512
1 − 0.512 = 0.488
∴ Decrease = 48.8% ✓
1 − 0.512 = 0.488
∴ Decrease = 48.8% ✓
Question 8
The longest diagonal of a cube is 9√3 cm. Find the total length of all the edges of the cube.
Explanation
The longest (space) diagonal of a cube of edge a is a√3.
a√3 = 9√3, so a = 9 cm.
A cube has 12 edges, so total length = 12a = 12 × 9 = 108 cm.
a√3 = 9√3, so a = 9 cm.
A cube has 12 edges, so total length = 12a = 12 × 9 = 108 cm.
Shortcut Method
Space diagonal = a√3 → a = 9
12 edges → 12 × 9
∴ Total = 108 cm ✓
12 edges → 12 × 9
∴ Total = 108 cm ✓
Question 9
Ramesh has a metal cube. He paints all sides of the cube with green color. He divides the cube into smaller cubes of volume 1 cu.m. each. How many of these smaller cubes will not have green color on even one of its sides if the volume of larger cube is 27 cu.m.?
Explanation
The volume of the big cube = 27 cu. m, so its edge = 3 m and it is divided into 27 unit cubes.
Only the cubes that are not on the surface have no green side. These form a 1 × 1 × 1 cube, so there is 1 such cube.
Only the cubes that are not on the surface have no green side. These form a 1 × 1 × 1 cube, so there is 1 such cube.
Question 10
Paper charge is Rs. 60 per kg. How much expenditure would be there to cover a cube of edge 10m with a paper, if one kg of paper covers 20 sq.m. area?
Explanation
The surface area of the cube = 6 × 10² = 600 sq. m.
The paper needed = 60020 = 30 kg, so the cost = 30 × 60 = Rs. 1800.
The paper needed = 60020 = 30 kg, so the cost = 30 × 60 = Rs. 1800.
Question 11
Sam has a solid metal ball with diameter 6cm. He melts it and uses the material for making a solid cylinder. If the diameter of the cylinder is same as the ball, what would its height be?
Explanation
The radius of the ball = 3 cm, so its volume = 43π × 3³ = 36π cu. cm.
The cylinder has radius 3 cm, so π × 3² × h = 36π and h = 4 cm.
The cylinder has radius 3 cm, so π × 3² × h = 36π and h = 4 cm.
Question 12
A steel vessel has a base of length 60 cm and breadth 30 cm. Water is poured in the vessel. A cubical steel box having edge of 30 cm is immersed completely in the vessel. How much will the water rise?
Explanation
The volume of the cube = 30³ = 27,000 cu. cm, which equals the volume of the water displaced.
The base area of the vessel = 60 × 30 = 1800 sq. cm.
Rise in the water level = 270001800 = 15 cm.
The base area of the vessel = 60 × 30 = 1800 sq. cm.
Rise in the water level = 270001800 = 15 cm.
Question 13
The surface area of a sphere is same as the curved surface area of a right circular cylinder whose height and diameter are 12cm each. The radius of the sphere is.
Explanation
The curved surface area of the cylinder = 2πrh = 2π × 6 × 12 = 144π sq. cm.
The surface area of the sphere = 4πR² = 144π, so R² = 36 and R = 6 cm.
The surface area of the sphere = 4πR² = 144π, so R² = 36 and R = 6 cm.
Question 14
If the areas of the three adjacent faces of a cuboidal box are 120cm2, 72cm2 and 60cm2 respectively, find the volume of the box.
Explanation
If l, b and h are the edges, then lb = 120, bh = 72 and hl = 60.
(lbh)² = 120 × 72 × 60 = 518,400, so the volume = lbh = 720 cu. cm.
(lbh)² = 120 × 72 × 60 = 518,400, so the volume = lbh = 720 cu. cm.
Question 15
Ramesh’s bedroom has a rectangular floor. He built the 4 walls of this room in Rs. 24000 at the rate of Rs. 20 per sq. m. What is height of his bedroom if perimeter of the floor is 150m?
Explanation
The area of the 4 walls = 2400020 = 1200 sq. m.
The area of the walls = perimeter × height = 150 × h, so h = 1200150 = 8 m.
The area of the walls = perimeter × height = 150 × h, so h = 1200150 = 8 m.
Question 16
Suresh wanted to keep volume of his right circular cone shaped cap same, but increase its height to 9 times. By how much should he increase or decrease the radius?
Explanation
The volume of a cone = 13πr²h. If the height becomes 9 times, then r² must become 19 of its value to keep the volume the same.
So r must be reduced to 13 of its value.
So r must be reduced to 13 of its value.
Question 17
What will be total cost of polishing curved surface of a wooden cylinder at rate of Rs. 20 per sq.m, if it has 40 cm diameter and 7m height?
Explanation
The radius = 20 cm = 0.2 m and the height = 7 m. The curved surface area = 2πrh = 2 × 227 × 0.2 × 7 = 8.8 sq. m.
Cost = 8.8 × 20 = Rs. 176.
Cost = 8.8 × 20 = Rs. 176.
Question 18
On decreasing each side of cube by 21%, its surface area decreases by?
Explanation
If each side decreases by 21%, the new side = 0.79 of the old side, so the new surface area = 0.79² = 0.6241 of the old one.
The surface area decreases by 100 - 62.41 = 37.59%.
The surface area decreases by 100 - 62.41 = 37.59%.
Question 19
Volume of a sphere with radius r is obtained by multiplying its surface area by r and a constant. Find the value of constant:
Explanation
The volume of a sphere = 43πr³ and its surface area = 4πr².
Volume = 13 × (4πr²) × r, so the constant = 13.
Volume = 13 × (4πr²) × r, so the constant = 13.
Question 20
The volume of a cube varies directly with the cube of its side. If 3 cubes of side 6 cm, 8 cm and 10 cm are melted together to form a fourth cube, find the side of the fourth cube.
Explanation
Total volume = 63+83+103 = 216+512+1000 = 1728 = 123, so the fourth cube's side is 12 cm.
Question 21
The heights of two cones are in the ratio 1:3. Their radii are in the ratio 3:1. Find the ratio of their volumes.
Explanation
Volume is proportional to r2h. Ratio = (32 × 1):(12 × 3) = 9:3 = 3:1.
Question 22
The curved surface area of a sphere is 5544 sq. cm. What is its volume?
Explanation
4πr2=5544 gives r2=441, so r=21 cm. Volume = (43)πr3 = (43)x(227)x9261 = 38808 cubic cm.
Question 23
Each edge of a cube is increased by 50%. What is the percentage change in surface area?
Explanation
New surface area/old = (1.5)2 = 2.25, a 125% increase.
Question 24
What is the ratio of the volumes of a right cylinder to that of a right cone having equal diameters and equal heights?
Explanation
For a cylinder and cone of equal radius and height: Volume of cylinder : Volume of cone = πr2h : 13πr2h = 3 : 1.
Question 25
What is the mass of the man, if a boat of length 4m, width 1.5m floating on a river sinks by 0.015 m when the man gets on the boat?
Explanation
Volume of water displaced = 4 × 1.5 × 0.015 = 0.09 m3. Mass = volume of water displaced × density of water = 0.09 × 1000 = 90 kg.
