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Year 5 Maths 3D Objects, Volume & Capacity

Volume with cubic units (build prisms from cubes)

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Theory

Volume is the amount of space an object takes up. We measure it in cubes: a cubic centimetre (\(1\,\text{cm}^3\)) is a cube with \(1\) cm edges. This Year 5 maths page shows how to find the volume of a prism built from cubes by counting one layer and multiplying by the number of layers.

Volume is the space an object takes up, measured in cubic units.

A cubic centimetre (\(\text{cm}^3\)) is the volume of a cube \(1\) cm long, \(1\) cm wide and \(1\) cm high. A cubic metre (\(\text{m}^3\)) is a cube with \(1\) m edges, used for large spaces.

A rectangular prism built from cubes is made of equal layers. Cubes in one layer \(=\) rows \(\times\) cubes in each row.

Volume \(=\) cubes in one layer \(\times\) number of layers. Count every cube, including the ones hidden at the back.

The same number of cubes can make prisms of different shapes. They all have the same volume.

Prism of centimetre cubes 5 by 3 by 4 A rectangular prism made of centimetre cubes, 5 cubes long, 3 cubes deep and 4 layers high. The top layer is shaded orange.
One layer: \(5 \times 3 = 15\) cubes. Four layers: \(15 \times 4 = 60\,\text{cm}^3\).
Top view of one layer A grid 6 squares across and 3 squares down, all shaded. Each square is the top of one cube in the layer.
Top view of one layer: \(3\) rows of \(6\), so \(18\) cubes.
Steps made of cubes A model of three steps built from centimetre cubes, 3 cubes deep. From left to right the columns are 3, 2 and 1 cubes high.
Steps \(3\) deep: front face \(3 + 2 + 1 = 6\) cubes, so \(6 \times 3 = 18\,\text{cm}^3\).

Counting cubes in a prism:

\[ \text{cubes in a layer} = \text{rows} \times \text{cubes in a row} \]
cubes in a layer = rows × cubes in a row
\[ \text{volume} = \text{cubes in a layer} \times \text{layers} \]
volume = cubes in a layer × layers
\[ 5 \times 3 \times 4 = 60\,\text{cm}^3 \]
5×3×4=60 cm
Cubic, not square. Volume is in \(\text{cm}^3\); area is in \(\text{cm}^2\).
Hidden cubes count. A solid prism has cubes behind the ones you can see. Use layers instead of counting faces.

Volume of a prism of cubes

  1. Find the cubes in one layer: rows \(\times\) cubes in each row.
  2. Count the layers.
  3. Multiply and write the answer in \(\text{cm}^3\).

All prisms from a number of cubes

  1. List sets of three whole numbers that multiply to the number of cubes.
  2. Start with \(1 \times 1 \times\) the number, then try \(1 \times 2\), \(1 \times 3\), and so on.
  3. Skip prisms that are just turned around.
Example 1 — One layer
Use the second diagram. What is the volume of the layer?
Solution
\(6 \times 3\)\(=\)\(18\,\text{cm}^3\)
6×3=18 cm

\(18\,\text{cm}^3\).

Example 2 — Layers
Use the first diagram. Find the volume of the prism.
Solution
\(5 \times 3\)\(=\)\(15\)
\(15 \times 4\)\(=\)\(60\,\text{cm}^3\)
5×3×4=60 cm

\(60\,\text{cm}^3\).

Example 3 — Take a layer away
The top layer of the first prism is removed. What is the new volume?
Solution
\(60 - 15\)\(=\)\(45\,\text{cm}^3\)
60−15=45 cm

\(45\,\text{cm}^3\) (three layers of \(15\)).

Example 4 — Steps
Use the third diagram. Find the volume of the steps.
Solution
\(3 + 2 + 1\)\(=\)\(6\)
\(6 \times 3\)\(=\)\(18\,\text{cm}^3\)
(3+2+1)×3=18 cm

\(18\,\text{cm}^3\).

Example 5 — Same cubes, different prisms
Find every rectangular prism that uses exactly \(18\) centimetre cubes.
Solution
\(1 \times 1 \times 18\)\(=\)\(18\)
\(1 \times 2 \times 9\)\(=\)\(18\)
\(1 \times 3 \times 6\)\(=\)\(18\)
\(2 \times 3 \times 3\)\(=\)\(18\)
1×1×18=1×2×9=1×3×6=2×3×3=18

\(4\) prisms, each with a volume of \(18\,\text{cm}^3\).

Example 6 — Layers from the volume
A box holds \(40\) cubes in full layers. Each layer is \(4\) by \(2\). How many layers are there?
Solution
\(4 \times 2\)\(=\)\(8\)
\(40 \div 8\)\(=\)\(5\)
40÷8=5

\(5\) layers.

Example 7 — Choose a unit
Which unit, \(\text{cm}^3\) or \(\text{m}^3\), would you use for
(i) an eraser?
(ii) a swimming pool?
Solution

(i) \(\text{cm}^3\)
(ii) \(\text{m}^3\)

Example 8 — Compare
Prism A is \(6\) by \(4\) by \(1\). Prism B is \(7\) by \(2\) by \(1\).
(i) Find each volume.
(ii) Which is bigger, and by how much?
Solution
\(6 \times 4 \times 1\)\(=\)\(24\)
\(7 \times 2 \times 1\)\(=\)\(14\)
6×4×1=24,7×2×1=14

(i) A: \(24\,\text{cm}^3\), B: \(14\,\text{cm}^3\)
(ii) A, by \(10\,\text{cm}^3\)

Common pitfalls

Counting only the cubes you can see. Hidden cubes at the back and inside still take up space.
Using square units. Volume is in cubic units such as \(\text{cm}^3\).
Adding instead of multiplying the layers. Four layers of \(15\) is \(15 \times 4\), not \(15 + 4\).
Counting a turned prism twice. \(2 \times 3 \times 3\) and \(3 \times 3 \times 2\) are the same prism turned around.

Frequently asked questions

What is volume?

The amount of space an object takes up.

What is a cubic centimetre?

The volume of a cube with \(1\) cm edges, written \(1\,\text{cm}^3\).

How do you find the volume of a prism made of cubes?

Find the cubes in one layer, then multiply by the number of layers: \(15 \times 4 = 60\,\text{cm}^3\).

Do hidden cubes count?

Yes. Every cube in the prism is part of its volume.

Can different prisms have the same volume?

Yes. \(1 \times 2 \times 9\) and \(2 \times 3 \times 3\) both use \(18\) cubes.

When do you use cubic metres?

For big spaces such as rooms, sheds and shipping containers.

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