Metric Units for Volume
Cubic units change by the length factor cubed, so \(1\text{ cm}^3 = 10 \times 10 \times 10 = 1000\text{ mm}^3\). This NSW Year 8 maths subtopic (NESA outcome MA4‑VOL‑C‑01) joins space and capacity: \(1\text{ cm}^3\) holds 1 mL, and \(1\text{ m}^3\) holds 1000 L, which is 1 kilolitre.
You will convert between \(\text{mm}^3\), \(\text{cm}^3\) and \(\text{m}^3\) and between mL, L, kL and megalitres, one step at a time, and change amounts to one unit before comparing them. You will also count the full doses in a medicine bottle, cost filling a school pool by the kilolitre, total a year of shower water and work out how long a town's dam will last. It is part of Year 8 Volume.
Every question with a fully worked solution.
- Metric Units for Volume - Video - Metric Units for Volume 1 Watch
Theory
Volume is the space a solid takes up; capacity is how much a container holds. Cubic units jump by the length factor cubed: \(1~\text{cm}^3 = 10 \times 10 \times 10 = 1000~\text{mm}^3\). And a space of \(1~\text{cm}^3\) holds exactly \(1\) mL. This Year 8 maths page shows how.
Cube the length factor. \(1\) cm \(= 10\) mm, so \(1~\text{cm}^3 = 10 \times 10 \times 10 = 1000~\text{mm}^3\). \(1\) m \(= 100\) cm, so \(1~\text{m}^3 = 100 \times 100 \times 100 = 1\,000\,000~\text{cm}^3\).
Capacity units. \(1\) L \(= 1000\) mL; \(1\) kilolitre (kL) \(= 1000\) L; \(1\) megalitre (ML) \(= 1000\) kL.
The link. \(1~\text{cm}^3 = 1\) mL, so \(1000~\text{cm}^3 = 1\) L and \(1~\text{m}^3 = 1000\) L \(= 1\) kL.
Which way? To a smaller unit, multiply (more of them). To a larger unit, divide. Take two steps one at a time along the chain.
Choose a sensible unit: mL for a dose of medicine, L for a bucket, kL for a water bill, ML for a dam. Compare, add or divide only in the same unit.
Rates and costs. Amount \(=\) rate \(\times\) time; time \(=\) amount \(\div\) rate. A water bill charges per kilolitre.
Volume and capacity:
| From | To | Do |
|---|---|---|
| cm\(^3\) | mm\(^3\) | \(\times 1000\) |
| m\(^3\) | cm\(^3\) | \(\times 1\,000\,000\) |
| cm\(^3\) | mL | same number |
| L | mL | \(\times 1000\) |
| kL | L | \(\times 1000\) |
| ML | kL | \(\times 1000\) |
| m\(^3\) | kL | same number |
How to convert
- Find the factor (cube the length factor, or use the capacity chain).
- Multiply to a smaller unit; divide to a larger unit.
- Check the size of the answer makes sense.
| \(2.4 \times 1\,000\,000\) | \(=\) | \(2\,400\,000\) |
\(2\,400\,000~\text{cm}^3\)
| \(3400 \div 1000\) | \(=\) | \(3.4\) |
\(3.4~\text{cm}^3\), since \(1000~\text{mm}^3 = 1~\text{cm}^3\).
(ii) Convert \(2\,300\,000~\text{cm}^3\) to m\(^3\).
| \(12.5 \times 1000\) | \(=\) | \(12\,500\) |
| \(2\,300\,000 \div 1\,000\,000\) | \(=\) | \(2.3\) |
(i) \(12\,500~\text{mm}^3\)
(ii) \(2.3~\text{m}^3\)
(i) Find its volume.
(ii) How many litres does it hold?
| \(V\) | \(=\) | \(35 \times 30 \times 20\) |
| \(\) | \(=\) | \(21\,000\) |
| \(21\,000 \div 1000\) | \(=\) | \(21\) |
(i) \(21\,000~\text{cm}^3\)
(ii) \(21\,000\) mL \(= 21\) L
| \(0.9 \times 1000\) | \(=\) | \(900~\text{mL}\) |
| \(0.0008 \times 1000\) | \(=\) | \(0.8~\text{L}\) |
| \(0.8 \times 1000\) | \(=\) | \(800~\text{mL}\) |
\(0.0008~\text{m}^3\), \(850\) mL, \(0.9\) L
(i) How many kilolitres is that?
(ii) How many cubic metres?
(iii) A pump moves \(25\) kL each hour. How long does it take to empty the full dam?
| \(0.8 \times 1000\) | \(=\) | \(800~\text{kL}\) |
| \(800 \div 25\) | \(=\) | \(32\) |
(i) \(800\) kL
(ii) \(800~\text{m}^3\) (\(1\) kL \(= 1~\text{m}^3\))
(iii) \(32\) hours
Common pitfalls
Frequently asked questions
Why do cubic units jump so much?
Each edge changes by the length factor, so the volume changes by it three times: \(10 \times 10 \times 10\) from cm\(^3\) to mm\(^3\).
How many cubic centimetres are in a cubic metre?
\(1\,000\,000\).
How many millilitres does 1 cm³ hold?
Exactly \(1\) mL.
How many litres are in a cubic metre?
\(1000\) L, which is \(1\) kL.
What is a megalitre?
\(1000\) kL, or \(1\,000\,000\) L.