Metric Units for Volume
Volume is the space a solid takes up, measured in cubic units such as \(\text{mm}^3\), \(\text{cm}^3\) and \(\text{m}^3\). This NSW Year 7 maths subtopic (NESA outcome MA4‑VOL‑C‑01) builds each conversion from a cube: a 1 cm cube is 10 mm along each edge, so \(1\text{ cm}^3 = 1000\text{ mm}^3\).
You will convert between cubic units by multiplying or dividing by 1000 or 1 000 000, avoid the trap of using the length fact for cubes, choose a sensible unit for a sugar cube or a shipping container, and work out how many bags of sand fill a sandpit, then compare volumes written in different units. It is part of Year 7 Area and Volume.
Every question with a fully worked solution.
- Metric Units for Volume - Video - Converting Metric Units of Volume Watch
Theory
The volume of a solid is the amount of space it takes up, measured in cubic units. A \(1\) cm cube is \(10\) mm long, wide and high, so \(1~\text{cm}^3 = 10 \times 10 \times 10~\text{mm}^3 = 1000~\text{mm}^3\). This Year 7 maths page gives the volume conversions, the chart and how to choose a sensible unit.
\(1~\text{cm}^3\) (one cubic centimetre) is the volume of a cube with \(1\) cm edges: \(1~\text{cm}^3 = 1~\text{cm} \times 1~\text{cm} \times 1~\text{cm}\).
Since \(1\) cm \(= 10\) mm, that cube holds \(10 \times 10 = 100\) tiny cubes in each layer and has \(10\) layers: \(1~\text{cm}^3 = 10 \times 10 \times 10~\text{mm}^3 = 1000~\text{mm}^3\).
A cube with \(1\) m edges has \(100\) cm edges: \(1~\text{m}^3 = 100 \times 100 \times 100~\text{cm}^3 = 1\,000\,000~\text{cm}^3\).
Larger unit to smaller unit: multiply (more of them). Smaller unit to larger unit: divide (fewer of them).
Choose a sensible unit: mm\(^3\) for tiny things (a grain of sand, a bead), cm\(^3\) for things that fit in your hand (a die, a sugar cube), m\(^3\) for big spaces (a room, a skip bin, a shipping container). Capacity units (mL, L, kL) are on the Volume and Capacity page.
Volume facts:
How to convert units of volume
- Find both units on the chart.
- Multiply to go to a smaller unit; divide to go to a larger unit.
- Check: a smaller unit gives a bigger number.
| \(24 \times 1000\) | \(=\) | \(24\,000\) |
\(24\,000~\text{mm}^3\)
| \(3400 \div 1\,000\,000\) | \(=\) | \(0.0034\) |
\(1~\text{cm}^3 = \dfrac{1}{1\,000\,000}~\text{m}^3\), so \(3400~\text{cm}^3 = 0.0034~\text{m}^3\).
(i) a die
(ii) a skip bin
(iii) a sesame seed
(i) cm\(^3\) (about \(1.5 \times 1.5 \times 1.5 \approx 3.4~\text{cm}^3\))
(ii) m\(^3\) (about \(3 \times 2 \times 1 = 6~\text{m}^3\))
(iii) mm\(^3\)
(ii) Convert \(6500~\text{mm}^3\) to cm\(^3\).
| \(8 \times 1000\) | \(=\) | \(8000\) |
| \(6500 \div 1000\) | \(=\) | \(6.5\) |
(i) \(8000~\text{mm}^3\)
(ii) \(6.5~\text{cm}^3\)
| \(0.3 \times 1\,000\,000\) | \(=\) | \(300\,000\) |
In cm\(^3\): \(300\,000 > 45\,000\), so \(0.3~\text{m}^3\) is larger. (Or \(45\,000~\text{cm}^3 = 0.045~\text{m}^3\).)
| \(3.5 \times 1\,000\,000\) | \(=\) | \(3\,500\,000\) |
| \(3\,500\,000 \div 7000\) | \(=\) | \(500\) |
\(500\) buckets
| \(100 \div 20\) | \(=\) | \(5\) |
| \(5 \times 5 \times 5\) | \(=\) | \(125\) |
\(5\) fit along each edge, so \(125\) small cubes.
Common pitfalls
Frequently asked questions
How many cubic millimetres are in a cubic centimetre?
\(1000\): \(10 \times 10 \times 10\).
How many cubic centimetres are in a cubic metre?
\(1\,000\,000\): \(100 \times 100 \times 100\).
How do you change cm³ to m³?
Divide by \(1\,000\,000\).
Which unit suits a shipping container?
Cubic metres (m\(^3\)).
Why is the volume factor so big?
Three lengths are multiplied, so the length factor is used three times.