Cube and Cuboid: Volume, Surface Area and Diagonal

A cuboid is a box with six rectangular faces, length l, breadth b and height h. A cube is a cuboid whose edges are all equal to a. Every question in this chapter uses one of a handful of formulas.

Quantity	Cuboid	Cube
Volume	l × b × h	a³
Total surface area	2(lb + bh + hl)	6a²
Lateral surface area (four walls)	2h(l + b)	4a²
Space diagonal	√(l² + b² + h²)	a√3

Volume is measured in cubic units, such as cm³, and areas in square units, such as cm². Options often include the same number with the wrong unit, so check which one the question asks for.

Worked example 1: volume of a box

A box measures 12 cm × 8 cm × 5 cm. Its volume is 12 × 8 × 5 = 480 cm³.

Its total surface area is 2(96 + 40 + 60) = 392 cm².

The space diagonal runs from one corner to the opposite corner through the inside of the box. It is the longest straight rod that fits inside.

Worked example 2: the diagonal

For a 12 × 4 × 3 box, the diagonal is √(144 + 16 + 9) = √169 = 13.

For a cube, one known measurement fixes everything, because the side a determines the rest. Work back to the side first, then compute what the question asks for.

Worked example 3: from surface area to volume

A cube has total surface area 150 cm². Then 6a² = 150, so a² = 25 and a = 5. The volume is 5³ = 125 cm³.

Worked example 4: from the diagonal

A cube's diagonal is 6√3 cm. Since the diagonal is a√3, the side is 6 cm and the volume is 216 cm³.

Sometimes a question gives the areas of three faces that meet at a corner instead of the edges. These areas are lb, bh and hl. Multiplying them gives (lbh)², so:

Volume = √(product of the three face areas)

Worked example 5: areas 24, 32 and 48

V = √(24 × 32 × 48) = √36,864 = 192 cm³. The edges are 4, 6 and 8 cm, since 4 × 6 = 24, 4 × 8 = 32 and 6 × 8 = 48.

Painting the walls of a room uses the lateral surface area, because the floor and ceiling are usually excluded. Subtract doors and windows if the question gives them.

Worked example 6: painting four walls

A room is 10 m long, 8 m wide and 4 m high. The walls cover 2 × 4 × (10 + 8) = 144 m². At ₹20 per m², the cost is ₹2,880.

Capacity questions convert volume into litres. One cubic metre holds 1,000 litres, and one litre is 1,000 cm³. Convert all lengths to metres or all to centimetres before multiplying, or the answer will be off by a power of ten.

Worked example 7: a water tank

A tank is 2 m long, 1.5 m wide and 1 m deep. It holds 2 × 1.5 × 1 = 3 m³, which is 3,000 litres.

When solids are melted and recast, the volume stays the same. Add the volumes of the pieces, then find the new shape's dimension.

Worked example 8: three cubes into one

Cubes of side 3, 4 and 5 cm are melted into one cube. Its volume is 27 + 64 + 125 = 216 cm³, so its side is 6 cm.

Scaling the side by a factor k multiplies areas by k² and volumes by k³. Use multipliers, not added percentages.

Worked example 9: a 10% longer side

Each edge of a cube grows by 10%. The volume is multiplied by 1.1³ = 1.331, an increase of 33.1%, while the surface area rises by 21%.

✘ Forgetting the 2 in 2(lb + bh + hl).
✘ Using total surface area for the four walls of a room.
✘ Mixing cm² and cm³ in the answer.
✘ Adding percentages instead of multiplying when the side changes.

Painted cubes cut into smaller cubes have their own counting method; see cube cutting and painting. Round solids are covered on cylinder, cone and sphere, and cube roots come quickly with the cube root trick.

Q1. What is the total surface area of a cuboid measuring 10 cm × 6 cm × 4 cm? easy

(a) 240 cm² (b) 248 cm² (c) 124 cm² (d) 240 cm³

✅ Answer: (b) 248 cm²

💡 Explanation: TSA = 2(lb + bh + hl) = 2(60 + 24 + 40) = 2 × 124 = 248 cm². Forgetting the factor 2 gives 124.

Q2. A cube has a volume of 343 cm³. What is its total surface area? easy

(a) 196 cm² (b) 245 cm² (c) 343 cm² (d) 294 cm²

✅ Answer: (d) 294 cm²

💡 Explanation: The side is ∛343 = 7 cm. TSA = 6a² = 6 × 49 = 294 cm². 196 cm² is the lateral surface area, 4a².

Q3. What is the length of the longest rod that fits in a box 6 m × 8 m × 24 m? medium

(a) 26 m (b) 24 m (c) 38 m (d) 28 m

✅ Answer: (a) 26 m

💡 Explanation: The longest rod lies along the space diagonal: √(36 + 64 + 576) = √676 = 26 m.

Q4. Three faces of a cuboid that meet at a corner have areas 12 cm², 15 cm² and 20 cm². What is its volume? medium

(a) 47 cm³ (b) 120 cm³ (c) 60 cm³ (d) 3,600 cm³

✅ Answer: (c) 60 cm³

💡 Explanation: The product of the three face areas is lb × bh × hl = (lbh)². So V = √(12 × 15 × 20) = √3,600 = 60 cm³.

Q5. How many cubes of side 2 cm can be cut from a cube of side 8 cm? easy

(a) 4 (b) 16 (c) 32 (d) 64

✅ Answer: (d) 64

💡 Explanation: Divide the volumes: 8³ ÷ 2³ = 512 ÷ 8 = 64. Equivalently, 4 small cubes fit along each edge, and 4³ = 64.

Q6. If each edge of a cube is increased by 20%, by what percentage does its surface area increase? medium

(a) 44% (b) 40% (c) 20% (d) 72.8%

✅ Answer: (a) 44%

💡 Explanation: Surface area depends on the square of the side: 1.2² = 1.44, an increase of 44%. The volume would rise by 1.2³ − 1 = 72.8%.

What is the difference between total and lateral surface area?

Total surface area counts all six faces. Lateral surface area counts only the four side faces, leaving out the top and bottom. For a room, the lateral area is the area of the four walls, which is what a painting question usually needs.

Why does the face-area trick work?

The three faces meeting at one corner have areas lb, bh and hl. Their product is l²b²h², which is the square of the volume lbh. Taking the square root of the product therefore gives the volume directly.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 16 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-23 against the sources listed above.

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