Volume & Surface Areas Questions and Answers

Q: How many smaller cubes are there having only one surface painted with red color ?

There are 54 cubes with only one face painted.A 15 cm cube cut into 3 cm cubes has \(15 / 3 = 5\) small cubes along each edge. Cubes with exactly one painted face lie in the middle of each face, away from the edges: a \(3 \times 3\) block on each face.\[ 6 \times (5 - 2)^{2} = 6 \times 9 = 54 \]The full breakdown of the 125 small cubes is: 8 corner cubes with three painted faces, \(12 \times 3 = 36\) edge cubes with two, 54 face cubes with one, and \(3^{3} = 27\) inner cubes with none; no cube has four or more painted faces. Check: \(8 + 36 + 54 + 27 = 125\).36 is the count of edge cubes (two painted faces), not one-face cubes, and 24 and 17 do not match any category.

Q: How many smaller cubes can be obtained from the larger cube ?

The larger cube gives 125 smaller cubes.With 5 small cubes along each edge, the total is:\[ 5^{3} = 125 \]Equivalently, the volume ratio is \(15^{3} / 3^{3} = 3375 / 27 = 125\). 25 counts only one layer (\(5^{2}\)), and 75 counts three layers.

Q: Water is flowing at the rate of 10 km/hr through a cylindrical pipe of diameter 14 cm into a rectangular tank which is 100 m long and 44 m wide. In what time the level of water in the tank will be risen by 7 cm ?

The level rises by 7 cm in 2 hours.The volume needed is the tank's base area times the rise in level, 7 cm or 0.07 m:\[ 100 \times 44 \times 0.07 = 308\ \text{m}^{3} \]The pipe has radius 7 cm, 0.07 m, and water moves 10 km, 10 000 m, in an hour, so the flow per hour is:\[ \begin{aligned} \pi r^{2} \times 10000 &= \tfrac{22}{7} \times 0.0049 \times 10000 \\ &= 154\ \text{m}^{3} \end{aligned} \]Time = \(308 \div 154 = 2\) hours. Using the diameter instead of the radius would give a flow four times larger and a time of half an hour.

Q: The cost of painting the surface of a solid cuboid of dimensions 40 cm × 36 cm × x cm at ₹ 20 per 100 cm2 is ₹ 1,032. What is the value of x ?

The value of x is 15.The cost is ₹ 20 per 100 square cm, so the area painted is:\[ \begin{aligned} \text{Area} &= \frac{1032}{20} \times 100 = 5160 \\ 2(40 \times 36 + 36x + 40x) &= 5160 \\ 1440 + 76x &= 2580 \\ x &= \frac{1140}{76} = 15\ \text{cm} \end{aligned} \]A check: 2(1440 + 540 + 600) = 5160 square cm, which costs 51.6 × 20 = ₹ 1,032. The other values give total surface areas of 5920 (x = 20), 6528 (x = 24) and 4704 (x = 12), none costing ₹ 1,032.

Q: The volume of a right circular cylinder of length 1·2 metres is 4620 cm3. What is the curved surface area (in cm2) of the cylinder ? (Take \(\pi = \dfrac{22}{7}\))

The curved surface area is 2640 square cm.The length is 1.2 m = 120 cm. From the volume:\[ \begin{aligned} \frac{22}{7} r^{2} \times 120 &= 4620 \\ r^{2} &= \frac{4620 \times 7}{22 \times 120} = 12{\cdot}25 \\ r &= 3{\cdot}5\ \text{cm} \\ \text{CSA} &= 2 \times \frac{22}{7} \times 3{\cdot}5 \times 120 = 2640 \end{aligned} \]The common error is to leave the length as 1.2 or 12 instead of converting it to 120 cm, which gives a wrong radius; the other options do not follow from a radius of 3.5 cm.

Q: How many planks of size 2 m × 35 cm × 8 cm can be made from a wooden block 8 m long, 70 cm wide and 40 cm thick ?

40 planks can be made.Put all lengths in centimetres. The block is 800 cm × 70 cm × 40 cm and each plank is 200 cm × 35 cm × 8 cm.\[ \begin{aligned} \text{planks} &= \frac{800 \times 70 \times 40}{200 \times 35 \times 8} \\ &= \frac{2240000}{56000} = 40 \end{aligned} \]The dimensions also fit exactly: 800/200 = 4 along the length, 70/35 = 2 across the width and 40/8 = 5 through the thickness, and 4 × 2 × 5 = 40, so no wood is wasted.Forgetting to convert metres to centimetres is the usual source of the other answers.

Q: A solid cube of side 12 cm is cut into small solid cubes of side 4 cm each. What will be the relation between the total surface area of the original cube and the new total surface area of the small cubes so formed ?

The new total surface area will be thrice the original.The big cube makes \((12/4)^3 = 27\) small cubes.\[ \begin{aligned} \text{original} &= 6 \times 12^2 = 864 \text{ cm}^2 \\ \text{new} &= 27 \times 6 \times 4^2 = 2592 \text{ cm}^2 \\ 2592 \div 864 &= 3 \end{aligned} \]Cutting creates new faces, so the total area increases, not decreases. In general, cutting into cubes of 1/n the side multiplies the surface area by n, here 3.The volume stays the same, 1728 cm3, because no material is lost; only the exposed surface grows.

Q: A metallic solid cuboidal box of dimensions \(200\text{ cm} \times 50\text{ cm} \times 100\text{ cm}\) is melted and recast into a solid cube. The difference between the surface areas of the two solids is

The difference is 10000 cm2.The volume stays the same when the box is recast:\[ \begin{aligned} V &= 200 \times 50 \times 100 = 10^{6}\ \text{cm}^{3} \\ \text{side} &= \sqrt[3]{10^{6}} = 100\ \text{cm} \\ S_{\text{cuboid}} &= 2(10000 + 5000 + 20000) \\ &= 70000 \\ S_{\text{cube}} &= 6 \times 100^{2} = 60000 \end{aligned} \]The difference is 70000 - 60000 = 10000 cm2. The cube has the smaller surface area, as expected: for a fixed volume, a cube encloses it with less surface than an elongated box.

Q: A godown is in the shape of a cuboid whose length, breadth and height are 56 m, 42 m and 10 m respectively. How many (maximum) cuboidal boxes each measuring 2.8 m × 2.5 m × 70 cm can be stored into the godown ?

The maximum number of boxes is 4800.Place each box with its 70 cm (0.7 m) side along the 56 m length, its 2.8 m side along the 42 m breadth and its 2.5 m side along the 10 m height:\[ \begin{aligned} 56 \div 0.7 &= 80 \\ 42 \div 2.8 &= 15 \\ 10 \div 2.5 &= 4 \\ 80 \times 15 \times 4 &= 4800 \end{aligned} \]Every division is exact, so the boxes fill the godown completely. The volume check agrees: \(56 \times 42 \times 10 = 23520\ \text{m}^{3}\) and each box is \(2.8 \times 2.5 \times 0.7 = 4.9\ \text{m}^{3}\), and 23520 ÷ 4.9 = 4800. Other orientations leave gaps and hold fewer boxes.

Q: The circumference of the base of a right circular cylinder is 528 cm and its height is 2 m. What is the volume of the cylinder ? \(\left(\text{Take } \pi = \frac{22}{7}\right)\)

The volume is 4.4352 m3.From the circumference, find the radius, then use \(V = \pi r^{2} h\) with h = 2 m = 200 cm:\[ \begin{aligned} 2\pi r &= 528 \\ r &= 528 \times \frac{7}{44} = 84\ \text{cm} \\ V &= \frac{22}{7} \times 84^{2} \times 200 \\ &= 4435200\ \text{cm}^{3} \end{aligned} \]Since 1 m3 = 1,000,000 cm3, V = 4.4352 m3. 2.2176 m3 is half the answer, as if the height were taken as 1 m; 6.6528 m3 is one and a half times it.

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If all the surfaces of a cube of 15 cm side are painted with red color and then it is cut into smaller cubes of 3 cm side, then answer the questions given below.

How many smaller cubes are there having only one surface painted with red color ?

A.17
B.24
C.36
D.54

Exam Year(s)

HPPSC HPAS - 2025,

Answer: D

There are 54 cubes with only one face painted.

A 15 cm cube cut into 3 cm cubes has \(15 / 3 = 5\) small cubes along each edge. Cubes with exactly one painted face lie in the middle of each face, away from the edges: a \(3 \times 3\) block on each face.

\[ 6 \times (5 - 2)^{2} = 6 \times 9 = 54 \]

The full breakdown of the 125 small cubes is: 8 corner cubes with three painted faces, \(12 \times 3 = 36\) edge cubes with two, 54 face cubes with one, and \(3^{3} = 27\) inner cubes with none; no cube has four or more painted faces. Check: \(8 + 36 + 54 + 27 = 125\).

36 is the count of edge cubes (two painted faces), not one-face cubes, and 24 and 17 do not match any category.

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How many smaller cubes can be obtained from the larger cube ?

A.25
B.35
C.75
D.125

Exam Year(s)

HPPSC HPAS - 2025,

Answer: D

The larger cube gives 125 smaller cubes.

With 5 small cubes along each edge, the total is:

\[ 5^{3} = 125 \]

Equivalently, the volume ratio is \(15^{3} / 3^{3} = 3375 / 27 = 125\). 25 counts only one layer (\(5^{2}\)), and 75 counts three layers.

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Water is flowing at the rate of 10 km/hr through a cylindrical pipe of diameter 14 cm into a rectangular tank which is 100 m long and 44 m wide. In what time the level of water in the tank will be risen by 7 cm ?

A.3 hours
B.4 hours
C.2 hours
D.1½ hours

Exam Year(s)

CBSE Assistant Secretary - 2024,

Answer: C

The level rises by 7 cm in 2 hours.

The volume needed is the tank's base area times the rise in level, 7 cm or 0.07 m:

\[ 100 \times 44 \times 0.07 = 308\ \text{m}^{3} \]

The pipe has radius 7 cm, 0.07 m, and water moves 10 km, 10 000 m, in an hour, so the flow per hour is:

\[ \begin{aligned} \pi r^{2} \times 10000 &= \tfrac{22}{7} \times 0.0049 \times 10000 \\ &= 154\ \text{m}^{3} \end{aligned} \]

Time = \(308 \div 154 = 2\) hours. Using the diameter instead of the radius would give a flow four times larger and a time of half an hour.

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The cost of painting the surface of a solid cuboid of dimensions 40 cm × 36 cm × x cm at ₹ 20 per 100 cm² is ₹ 1,032. What is the value of x ?

A.20
B.15
C.24
D.12

Exam Year(s)

CTET - 2023,

Answer: B

The value of x is 15.

The cost is ₹ 20 per 100 square cm, so the area painted is:

\[ \begin{aligned} \text{Area} &= \frac{1032}{20} \times 100 = 5160 \\ 2(40 \times 36 + 36x + 40x) &= 5160 \\ 1440 + 76x &= 2580 \\ x &= \frac{1140}{76} = 15\ \text{cm} \end{aligned} \]

A check: 2(1440 + 540 + 600) = 5160 square cm, which costs 51.6 × 20 = ₹ 1,032. The other values give total surface areas of 5920 (x = 20), 6528 (x = 24) and 4704 (x = 12), none costing ₹ 1,032.

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The volume of a right circular cylinder of length 1·2 metres is 4620 cm³. What is the curved surface area (in cm²) of the cylinder ? (Take \(\pi = \dfrac{22}{7}\))

A.2640
B.2200
C.2860
D.1980

Exam Year(s)

CTET - 2023,

Answer: A

The curved surface area is 2640 square cm.

The length is 1.2 m = 120 cm. From the volume:

\[ \begin{aligned} \frac{22}{7} r^{2} \times 120 &= 4620 \\ r^{2} &= \frac{4620 \times 7}{22 \times 120} = 12{\cdot}25 \\ r &= 3{\cdot}5\ \text{cm} \\ \text{CSA} &= 2 \times \frac{22}{7} \times 3{\cdot}5 \times 120 = 2640 \end{aligned} \]

The common error is to leave the length as 1.2 or 12 instead of converting it to 120 cm, which gives a wrong radius; the other options do not follow from a radius of 3.5 cm.

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How many planks of size 2 m × 35 cm × 8 cm can be made from a wooden block 8 m long, 70 cm wide and 40 cm thick ?

A.45
B.40
C.50
D.48

Exam Year(s)

CTET - 2024,

Answer: B

40 planks can be made.

Put all lengths in centimetres. The block is 800 cm × 70 cm × 40 cm and each plank is 200 cm × 35 cm × 8 cm.

\[ \begin{aligned} \text{planks} &= \frac{800 \times 70 \times 40}{200 \times 35 \times 8} \\ &= \frac{2240000}{56000} = 40 \end{aligned} \]

The dimensions also fit exactly: 800/200 = 4 along the length, 70/35 = 2 across the width and 40/8 = 5 through the thickness, and 4 × 2 × 5 = 40, so no wood is wasted.

Forgetting to convert metres to centimetres is the usual source of the other answers.

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A solid cube of side 12 cm is cut into small solid cubes of side 4 cm each. What will be the relation between the total surface area of the original cube and the new total surface area of the small cubes so formed ?

A.New total surface area will be thrice the original total surface area.
B.New total surface area will be \(\frac{1}{2}\) of the original total surface area.
C.New total surface area will be twice the original total surface area.
D.New total surface area will be \(\frac{1}{3}\) of the original total surface area.

Exam Year(s)

CTET - 2024,

Answer: A

The new total surface area will be thrice the original.

The big cube makes \((12/4)^3 = 27\) small cubes.

\[ \begin{aligned} \text{original} &= 6 \times 12^2 = 864 \text{ cm}^2 \\ \text{new} &= 27 \times 6 \times 4^2 = 2592 \text{ cm}^2 \\ 2592 \div 864 &= 3 \end{aligned} \]

Cutting creates new faces, so the total area increases, not decreases. In general, cutting into cubes of 1/n the side multiplies the surface area by n, here 3.

The volume stays the same, 1728 cm³, because no material is lost; only the exposed surface grows.

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A metallic solid cuboidal box of dimensions \(200\text{ cm} \times 50\text{ cm} \times 100\text{ cm}\) is melted and recast into a solid cube. The difference between the surface areas of the two solids is

A.\(25000\text{ cm}^{2}\)
B.\(20000\text{ cm}^{2}\)
C.\(15000\text{ cm}^{2}\)
D.\(10000\text{ cm}^{2}\)

Exam Year(s)

CTET - 2024,

Answer: D

The difference is 10000 cm².

The volume stays the same when the box is recast:

\[ \begin{aligned} V &= 200 \times 50 \times 100 = 10^{6}\ \text{cm}^{3} \\ \text{side} &= \sqrt[3]{10^{6}} = 100\ \text{cm} \\ S_{\text{cuboid}} &= 2(10000 + 5000 + 20000) \\ &= 70000 \\ S_{\text{cube}} &= 6 \times 100^{2} = 60000 \end{aligned} \]

The difference is 70000 - 60000 = 10000 cm². The cube has the smaller surface area, as expected: for a fixed volume, a cube encloses it with less surface than an elongated box.

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A godown is in the shape of a cuboid whose length, breadth and height are 56 m, 42 m and 10 m respectively. How many (maximum) cuboidal boxes each measuring 2.8 m × 2.5 m × 70 cm can be stored into the godown ?

A.2400
B.3600
C.4800
D.5400

Exam Year(s)

CTET - 2019,

Answer: C

The maximum number of boxes is 4800.

Place each box with its 70 cm (0.7 m) side along the 56 m length, its 2.8 m side along the 42 m breadth and its 2.5 m side along the 10 m height:

\[ \begin{aligned} 56 \div 0.7 &= 80 \\ 42 \div 2.8 &= 15 \\ 10 \div 2.5 &= 4 \\ 80 \times 15 \times 4 &= 4800 \end{aligned} \]

Every division is exact, so the boxes fill the godown completely. The volume check agrees: \(56 \times 42 \times 10 = 23520\ \text{m}^{3}\) and each box is \(2.8 \times 2.5 \times 0.7 = 4.9\ \text{m}^{3}\), and 23520 ÷ 4.9 = 4800. Other orientations leave gaps and hold fewer boxes.

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10.

The circumference of the base of a right circular cylinder is 528 cm and its height is 2 m. What is the volume of the cylinder ? \(\left(\text{Take } \pi = \frac{22}{7}\right)\)

A.\(2.2176\ \text{m}^{3}\)
B.\(3.3264\ \text{m}^{3}\)
C.\(4.4352\ \text{m}^{3}\)
D.\(6.6528\ \text{m}^{3}\)

Exam Year(s)

CTET - 2019,

Answer: C

The volume is 4.4352 m³.

From the circumference, find the radius, then use \(V = \pi r^{2} h\) with h = 2 m = 200 cm:

\[ \begin{aligned} 2\pi r &= 528 \\ r &= 528 \times \frac{7}{44} = 84\ \text{cm} \\ V &= \frac{22}{7} \times 84^{2} \times 200 \\ &= 4435200\ \text{cm}^{3} \end{aligned} \]

Since 1 m³ = 1,000,000 cm³, V = 4.4352 m³. 2.2176 m³ is half the answer, as if the height were taken as 1 m; 6.6528 m³ is one and a half times it.

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