The volume of a cuboid is the amount of three-dimensional space it occupies, measured in cubic units. The formula is length × width × height. A cuboid is any box-shaped solid with six rectangular faces — from a brick to a swimming pool — so this formula appears constantly across KS3 maths and science.
What is a cuboid?
A cuboid is a 3D solid with:
- 6 rectangular faces
- 12 edges
- 8 vertices (corners)
All the faces meet at right angles. A cube is a special cuboid where all three dimensions are equal. Common real-life cuboids include cereal boxes, shoeboxes, and rooms.
The volume formula
V = l × w × h
Where:
l= length (the longest horizontal dimension)w= width (the shorter horizontal dimension)h= height (the vertical dimension)
The three dimensions can be labelled in any order — multiplication is commutative — but it is good practice to identify length, width, and height clearly from a diagram.
Units of volume are always cubic (e.g. cm³, m³, mm³).
Why does length × width × height work?
Think of the base of the cuboid first. Its area is l × w. This tells you how many unit squares fit on the bottom layer. Each layer is 1 unit thick, and there are h layers. The total number of unit cubes is therefore l × w × h. For example, a cuboid that is 4 cm × 3 cm × 2 cm contains 4 × 3 = 12 cubes per layer, with 2 layers, giving 24 cubes — so the volume is 24 cm³.
Step-by-step method
- Identify the three dimensions from the diagram or question.
- Check that all dimensions share the same unit — convert if necessary.
- Multiply:
V = l × w × h. - Write the unit as a cubic unit (e.g. cm³, m³).
Worked examples — whole numbers
Worked example 1
Find the volume of a cuboid with length 5 cm, width 4 cm, and height 3 cm.
V = 5 × 4 × 3 = 60
Answer: 60 cm³
Worked example 2 — cube
A cube has side length 6 cm. Find its volume.
V = 6 × 6 × 6 = 216
Answer: 216 cm³
(A cube is a cuboid where l = w = h, so V = s³.)
Worked example 3 — real-life context
A fish tank is 80 cm long, 40 cm wide, and 50 cm tall. Find its volume.
V = 80 × 40 × 50 = 160 000
Answer: 160 000 cm³
This also equals 160 litres, since 1 litre = 1000 cm³. (160 000 ÷ 1000 = 160.)
Worked examples — decimal dimensions
Worked example 4
A box is 2.5 m long, 1.2 m wide, and 0.8 m high. Find its volume.
V = 2.5 × 1.2 × 0.8
2.5 × 1.2 = 3.0
3.0 × 0.8 = 2.4
Answer: 2.4 m³
Worked example 5
A chocolate bar has dimensions 15 cm × 7.5 cm × 2 cm. Find its volume.
V = 15 × 7.5 × 2
15 × 7.5 = 112.5
112.5 × 2 = 225
Answer: 225 cm³
Finding a missing dimension from the volume
Sometimes you are given the volume and two dimensions, and you must find the third.
Method: rearrange V = l × w × h to isolate the unknown dimension.
If V, l, and w are known: h = V ÷ (l × w).
Worked example 6
A cuboid has volume 180 cm³, length 9 cm, and width 5 cm. Find the height.
h = 180 ÷ (9 × 5) = 180 ÷ 45 = 4
Answer: height = 4 cm
Check: 9 × 5 × 4 = 180 cm³ ✓
Worked example 7
A rectangular fish pond has volume 12 m³. Its length is 4 m and its height (depth) is 0.75 m. Find the width.
w = 12 ÷ (4 × 0.75) = 12 ÷ 3 = 4
Answer: width = 4 m
Unit conversions for volume
Volume units grow very quickly because you are cubing a linear conversion factor.
| Conversion | Factor |
|---|---|
| 1 m = 100 cm | 1 m³ = 100³ cm³ = 1 000 000 cm³ |
| 1 cm = 10 mm | 1 cm³ = 10³ mm³ = 1000 mm³ |
| 1 litre = 1000 cm³ | 1 m³ = 1000 litres |
| 1 ml = 1 cm³ |
Common trap: students multiply by 100 when converting m³ to cm³. The correct factor is 100³ = 1 000 000.
Worked example 8 — unit conversion
A storage container has volume 0.006 m³. Convert this to cm³.
0.006 m³ × 1 000 000 = 6000 cm³
Answer: 6000 cm³
How volume of cuboids fits the KS3 national curriculum
The Department for Education's KS3 mathematics programme of study requires pupils to "derive and apply formulae to calculate and solve problems involving volume of cuboids (including cubes) and other prisms." BBC Bitesize's KS3 shape resources describe the cuboid volume formula as foundational for all 3D shape work, including the volume of prisms (which extends the method: area of cross-section × length) that appears in Year 9 and at GCSE.
Common mistakes
Mistake 1 — Adding the dimensions instead of multiplying.
5 + 4 + 3 = 12, but the volume of a 5 × 4 × 3 cuboid is 5 × 4 × 3 = 60 cm³. Always multiply all three.
Mistake 2 — Using incorrect units. If dimensions are in cm, volume is in cm³, not cm or cm². Write the cubic superscript.
Mistake 3 — Multiplying only two dimensions.
It is common to calculate the base area correctly (l × w) but then forget to multiply by the height. Volume requires all three dimensions.
Mistake 4 — Mixing units.
If one dimension is in m and the others are in cm, convert all to the same unit before multiplying. For example, a box that is 1 m × 40 cm × 20 cm: convert 1 m = 100 cm, then V = 100 × 40 × 20 = 80 000 cm³.
Volume vs surface area
Volume and surface area are different measures. Volume measures the space inside the cuboid; surface area measures the total area of all its outer faces.
For a cuboid with dimensions l, w, h:
- Volume:
V = lwh - Surface area:
SA = 2(lw + lh + wh)
On exam papers, both may be tested on the same diagram. Read the question carefully to identify which one is required.
Frequently asked questions
What is the difference between volume and capacity?
Volume is the total 3D space occupied by an object, including the material it is made of. Capacity is the amount a container can hold (its internal volume). In practice, KS3 problems treat the two as interchangeable for hollow containers such as tanks and boxes, since the wall thickness is typically ignored.
Does it matter which dimension I call length, width, and height?
No. Because V = l × w × h and multiplication is commutative, you will always get the same result. 5 × 4 × 3 = 4 × 5 × 3 = 3 × 4 × 5 = 60. What matters is that you identify three distinct dimensions and multiply all three.
How do I convert between cm³ and litres?
1 litre = 1000 cm³. To convert cm³ to litres, divide by 1000. To convert litres to cm³, multiply by 1000. So 2500 cm³ = 2.5 litres, and 3 litres = 3000 cm³. This conversion is frequently used in science and in real-life problems about tanks, pools, and bottles.
Can the volume formula be used for non-cuboid boxes?
Only if all the angles are right angles (i.e. it is truly a rectangular prism). If a box has slanted sides or irregular cross-sections, you cannot use l × w × h directly. At KS3 you extend to the prism formula V = A × l, where A is the area of the uniform cross-section and l is the length of the prism — but this is taught in Year 9.
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