Cube and Cuboid: Surface Area and Volume

Cube and cuboid are among the most important topics in Mensuration for competitive examinations such as JKSSB, SSC, Railway and other government exams. Questions are usually based on formulas, direct calculations, comparison, and statement-type problems.

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1. What is a Cuboid?

A cuboid is a three-dimensional solid having:

Length (l)

Breadth (b)
Height (h)

Examples: a book, brick, room, rectangular box.

A cuboid has:

6 faces

12 edges
8 vertices

Main formulas of Cuboid

Quantity	Formula
Total Surface Area (TSA)	2(lb + bh + hl)
Lateral Surface Area (LSA)	2h(l + b)
Volume	l × b × h
Area of four walls	2h(l + b)
Sum of all edges	4(l + b + h)
Diagonal	√(l² + b² + h²)

Teacher Explanation

Suppose a box has:

Length = 10 cm, Breadth = 5 cm, Height = 4 cm

Its volume is:

$$V=l\times b\times h$$ $$=10\times5\times4=200\ cm^3$$

So, the box can contain 200 cubic centimetres of space.

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2. What is a Cube?

A cube is a special type of cuboid in which:

$$l=b=h=a$$

where a = side of the cube.

Examples: dice, ice cube, Rubik's cube.

A cube has:

6 square faces

12 equal edges
8 vertices

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3. Important Formulas of Cube

Quantity	Formula
Total Surface Area	6a²
Lateral Surface Area	4a²
Volume	a³
Sum of all edges	12a
Diagonal of cube	a√3
Diagonal of one face	a√2

Where a = side of cube.

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4. Cube vs Cuboid — Important Comparison

Feature	Cube	Cuboid
Length	Equal to side	May be different
Breadth	Equal to side	May be different
Height	Equal to side	May be different
Faces	6	6
Edges	12	12
Vertices	8	8
Shape of faces	Squares	Rectangles
TSA	6a²	2(lb + bh + hl)
LSA	4a²	2h(l+b)
Volume	a³	lbh
Diagonal	a√3	√(l²+b²+h²)

⭐ Remember

Every cube is a cuboid, but every cuboid is NOT a cube.

Why?

Because a cube has all three dimensions equal, while a cuboid can have different length, breadth and height.

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5. Surface Area vs Volume

This is a very important distinction for examinations.

Surface Area

Surface area tells us about the outer covering of a solid.

Unit:

cm²

m²
km²

Volume

Volume tells us about the space occupied or capacity of a solid.

Unit:

cm³

m³
litres

Teacher Trick

Think of a water tank.

If you want to paint the outside → Surface Area

If you want to know how much water it can hold → Volume

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6. Example: Cube

A cube has side 5 cm.

Total Surface Area

$$TSA=6a^2$$ $$=6(5)^2$$ $$=150\ cm^2$$

Lateral Surface Area

$$LSA=4a^2$$ $$=4(25)=100\ cm^2$$

Volume

$$V=a^3$$ $$=5^3=125\ cm^3$$

Answer

TSA = 150 cm²

LSA = 100 cm²
Volume = 125 cm³

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7. Example: Cuboid

A cuboid has:

Length = 8 cm

Breadth = 5 cm
Height = 3 cm

TSA

$$2(lb+bh+hl)$$ $$=2(8\times5+5\times3+3\times8)$$ $$=2(40+15+24)$$ $$=158\ cm^2$$

Volume

$$lbh=8\times5\times3$$ $$=120\ cm^3$$

Therefore:

TSA = 158 cm²

Volume = 120 cm³

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8. Comparison-Based Concept

Suppose a cube and cuboid have the same volume.

Do their surface areas necessarily have to be equal?

No.

Volume depends on:

$$l\times b\times h$$

while surface area depends on:

$$2(lb+bh+hl)$$

Therefore, two solids can have the same volume but different surface areas.

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9. Important Exam Tricks

Trick 1: If side of cube is doubled

Original volume:

$$V=a^3$$

New volume:

$$(2a)^3=8a^3$$

Therefore, volume becomes 8 times.

But surface area:

$$6(2a)^2=24a^2$$

Therefore, surface area becomes 4 times.

⭐ Result

If side is doubled:

• Surface area → 4 times

• Volume → 8 times

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Trick 2: If side becomes three times

• Surface area → 9 times

• Volume → 27 times

Because:

$$3^2=9$$

and

$$3^3=27$$

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10. Unit Conversion

Remember:

$$1m=100cm$$

Therefore:

$$1m^2=10,000cm^2$$

and

$$1m^3=1,000,000cm^3$$

Also:

$$1 litre=1000cm^3$$

Very Important

Do not confuse square units and cubic units.

Surface area → square units

Volume → cubic units

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MCQs — Cube and Cuboid

Q1. A cube has side 4 cm. Its volume is:

A) 16 cm³
B) 32 cm³
C) 64 cm³
D) 96 cm³

Answer: C) 64 cm³

Q2. The total surface area of a cube of side 5 cm is:

A) 25 cm²
B) 100 cm²
C) 125 cm²
D) 150 cm²

Answer: D) 150 cm²

Q3. The lateral surface area of a cube of side 7 cm is:

A) 49 cm²
B) 98 cm²
C) 196 cm²
D) 294 cm²

Answer: C) 196 cm²

Q4. A cuboid has length 10 cm, breadth 5 cm and height 2 cm. Its volume is:

A) 50 cm³
B) 100 cm³
C) 150 cm³
D) 200 cm³

Answer: B) 100 cm³

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Q5. The formula for total surface area of a cuboid is:

A) lbh
B) 2(lb + bh + hl)
C) 2h(l+b)
D) 6a²

Answer: B) 2(lb + bh + hl)

Q6. Which of the following is the formula for the volume of a cube?

A) 6a²
B) 4a²
C) a³
D) 3a²

Answer: C) a³

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Q7. A cube has how many edges?

A) 6
B) 8
C) 10
D) 12

Answer: D) 12

Q8. A cuboid has how many vertices?

A) 6
B) 8
C) 10
D) 12

Answer: B) 8

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Q9. If the side of a cube is doubled, its volume becomes:

A) 2 times
B) 4 times
C) 6 times
D) 8 times

Answer: D) 8 times

Q10. If the side of a cube is doubled, its surface area becomes:

A) 2 times
B) 4 times
C) 6 times
D) 8 times

Answer: B) 4 times

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Q11. Which of the following is measured in cubic units?

A) Surface area
B) Perimeter
C) Volume
D) Length

Answer: C) Volume

Q12. Which of the following is measured in square units?

A) Volume
B) Surface area
C) Capacity
D) Mass

Answer: B) Surface area

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Q13. A cube has a side of 10 cm. Its lateral surface area is:

A) 100 cm²
B) 200 cm²
C) 400 cm²
D) 600 cm²

Answer: C) 400 cm²

Q14. A cuboid has dimensions 6 cm × 4 cm × 3 cm. Its volume is:

A) 24 cm³
B) 48 cm³
C) 72 cm³
D) 96 cm³

Answer: C) 72 cm³

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Q15. Every cube is:

A) A cylinder
B) A sphere
C) A cuboid
D) A cone

Answer: C) A cuboid

Statement-Type MCQs

Q16. Consider the following statements:

1. A cube has 6 faces.

2. A cube has 12 edges.

3. A cube has 8 vertices.

4. All faces of a cube are squares.

Which statements are correct?

A) 1 and 2 only
B) 1, 2 and 3 only
C) 2, 3 and 4 only
D) 1, 2, 3 and 4

Answer: D) 1, 2, 3 and 4

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Q17. Consider the following statements regarding a cuboid:

1. It has 6 faces.

2. It has 12 edges.

3. It has 8 vertices.

4. All its dimensions must be equal.

Which statement is incorrect?

A) 1
B) 2
C) 3
D) 4

Answer: D) 4

Explanation: A cuboid does not require all dimensions to be equal.

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Q18. Consider the following statements:

1. TSA of a cube = 6a².

2. LSA of a cube = 4a².

3. Volume of a cube = a³.

Which are correct?

A) 1 only
B) 1 and 2 only
C) 2 and 3 only
D) 1, 2 and 3

Answer: D) 1, 2 and 3

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Q19. If the side of a cube is increased from a to 2a:

1. Surface area becomes 4 times.

2. Volume becomes 8 times.

3. Edge length becomes 2 times.

Which is correct?

A) 1 only
B) 1 and 2 only
C) 2 and 3 only
D) 1, 2 and 3

Answer: D) 1, 2 and 3

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Q20. Consider the following:

Statement I: A cube is a special type of cuboid.

Statement II: In a cube, length, breadth and height are equal.

A) Both statements are correct
B) Both statements are incorrect
C) Statement I correct, Statement II incorrect
D) Statement I incorrect, Statement II correct

Answer: A) Both statements are correct

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⭐ 

🔥 Most Important Comparison

Change in cube	Surface Area	Volume
Side ×2	×4	×8
Side ×3	×9	×27
Side ×4	×16	×64
Side ×5	×25	×125

Exam Mantra:
👉 Surface Area → square of dimension
👉 Volume → cube of dimension
👉 Cube → all dimensions equal
👉 Cuboid → dimensions may be different