Surface Area and Volume: Complete Guide

Volume and surface area for every 3D object in the syllabus — cubes, rectangular prisms and triangular prisms — plus capacity, unit conversions and working backwards, each worked in full, with a worksheet.

For 3D objects, volume measures the space inside (cubic units like cm³) and surface area measures the total area of all the faces added together (square units like cm²). This guide covers every object type an exam sets, plus capacity and reverse questions.

Prism volume = area of cross-section × length • Rectangular prism V = l × b × h

A rectangular prism (box): volume = length × breadth × height; surface area adds all six faces.

1Type 1: Cubes

Worked ExampleExample 1: a cube with side 4 cm

1
Volume = side³ = 4 × 4 × 4 = 64 cm³.
All three dimensions are equal in a cube.
2
Surface area = 6 × (side²) = 6 × 16 = 96 cm².
A cube has 6 identical square faces.

2Type 2: Rectangular prisms (boxes)

Worked ExampleExample 2: a box 5 cm × 4 cm × 3 cm

1
Volume = 5 × 4 × 3 = 60 cm³.
Multiply the three dimensions.
2
The three different faces: (5×4) + (5×3) + (4×3) = 20 + 15 + 12 = 47.
One from each pair of opposite faces.
3
Surface area = 2 × 47 = 94 cm².
Each face has a matching opposite, so double the total.

3Type 3: Triangular prisms

Volume of any prism = area of the cross-section × length. For a triangular prism, find the triangle's area first.

Worked ExampleExample 3: triangular cross-section (base 6, height 4), prism length 10 cm

1
Cross-section area = 12 × 6 × 4 = 12 cm².
Area of the triangle end.
2
Volume = 12 × 10 = 120 cm³.
Cross-section area × length.

4Type 4: Capacity and unit conversions

Capacity is how much liquid an object holds — it comes from volume: 1 cm³ = 1 mℓ and 1 m³ = 1 000 ℓ = 1 kℓ.

Worked ExampleExample 4: how many millilitres does a 5 × 4 × 3 cm box hold?

1
Volume = 60 cm³ (from Example 2).
First find the volume.
2
Capacity = 60 mℓ.
1 cm³ = 1 mℓ.

5Type 5: Working backwards

Worked ExampleExample 5: a box has volume 84 cm³, base 7 cm × 3 cm. Find its height

1
Volume = l × b × h, so 84 = 7 × 3 × h = 21h.
Substitute the known values.
2
h = 84 ÷ 21 = 4 cm.
Divide the volume by the base area.

💡

Check your units. Volume is cubic (cm³, m³); surface area is square (cm², m²); capacity is mℓ, ℓ or kℓ. Mixing them up is a common lost mark.

⚠️

Don't confuse volume and surface area. 'How much water fits' → volume; 'how much paint/wrapping to cover it' → surface area.

Measurement is reliable, formula-driven marks. Practise every object type and the reverse questions below.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Volume of a cube with side 4 cm (cube)
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4³ = 64 cm³
2
Surface area of a cube with side 3 cm (cube)
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6 × 9 = 54 cm²
3
Volume of a box 6 × 3 × 2 cm (box)
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6 × 3 × 2 = 36 cm³
4
Surface area of a box 5 × 4 × 2 cm (box)
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2[(5×4)+(5×2)+(4×2)] = 2(38) = 76 cm²
5
Volume of a box 10 × 5 × 4 cm (box)
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200 cm³
6
Triangular prism: cross-section area 15 cm², length 8 cm. Volume? (prism)
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15 × 8 = 120 cm³
7
Triangular prism: triangle base 8, height 5, length 10 cm. Volume? (prism)
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12×8×5 = 20; 20×10 = 200 cm³
8
How many mℓ does a 200 cm³ container hold? (capacity)
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200 mℓ
9
A box has volume 60 cm³, base 5 × 4 cm. Find the height. (reverse)
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60 ÷ 20 = 3 cm
10
A cube has volume 125 cm³. Find its side length. (reverse)
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125 = 5 cm