Metric Units
Metric prefixes like kilo, centi and milli, and micro and nano for tiny lengths, link units by powers of \(10\). This NSW Year 9 Advanced maths subtopic converts units of length, area, volume, capacity and mass (NESA outcome MA5‑MAG‑C‑01).
You will convert between units such as km and mm, \(\text{m}^2\) and \(\text{cm}^2\), and \(\text{m}^3\) and litres, using the square or cube of the factor for area and volume. You will find the capacity of a pool and a tank, show that \(1\ \text{m}^3\) of water is \(1\) kL and \(1\) tonne, find the hectares and fertiliser for a paddock, and find the mass of a gold bar. It is part of Year 9 Advanced Surface Area and Volume.
Theory
The metric system uses powers of \(10\), so converting is multiplying or dividing. The catch is that area factors are squared and volume factors are cubed. This Year 9 Advanced maths page covers every conversion, the link between volume and capacity, and prefixes from nano to tera.
Metric units are built from a base unit (metre, gram, litre) and a prefix that tells you the power of \(10\): kilo means \(1000\), centi means \(\dfrac{1}{100}\), milli means \(\dfrac{1}{1000}\).
To convert to a smaller unit, multiply (you need more of them). To convert to a larger unit, divide.
Area units are squares. A square metre is \(100\) cm by \(100\) cm, so \(1~\text{m}^2 = 10\,000~\text{cm}^2\). Volume units are cubes, so the factor is cubed: \(1~\text{m}^3 = 100^3 = 1\,000\,000~\text{cm}^3\).
Capacity is how much a container holds. It links to volume: \(1~\text{cm}^3 = 1~\text{mL}\) and \(1~\text{m}^3 = 1000~\text{L} = 1~\text{kL}\). For water, \(1\) mL has a mass of \(1\) g, so \(1~\text{m}^3\) of water has a mass of \(1\) tonne.
Very small and very large measurements use prefixes such as micro (\(\mu\), a millionth), nano (a billionth), giga (a billion) and tera (a trillion), often written in scientific notation.
Conversion factors:
| Quantity | Units and links |
|---|---|
| Length | \(1~\text{km} = 1000~\text{m}\), \(1~\text{m} = 100~\text{cm}\), \(1~\text{cm} = 10~\text{mm}\) |
| Area | \(1~\text{cm}^2 = 100~\text{mm}^2\), \(1~\text{m}^2 = 10\,000~\text{cm}^2\), \(1~\text{ha} = 10\,000~\text{m}^2\), \(1~\text{km}^2 = 100~\text{ha}\) |
| Volume | \(1~\text{cm}^3 = 1000~\text{mm}^3\), \(1~\text{m}^3 = 1\,000\,000~\text{cm}^3\) |
| Capacity | \(1~\text{L} = 1000~\text{mL}\), \(1~\text{kL} = 1000~\text{L}\), \(1~\text{cm}^3 = 1~\text{mL}\), \(1~\text{m}^3 = 1~\text{kL}\) |
| Mass | \(1~\text{kg} = 1000~\text{g}\), \(1~\text{t} = 1000~\text{kg}\); \(1\) mL of water has a mass of \(1\) g |
Common prefixes:
| Prefix | Symbol | Meaning |
|---|---|---|
| tera | T | \(10^{12}\) |
| giga | G | \(10^{9}\) |
| mega | M | \(10^{6}\) |
| kilo | k | \(10^{3}\) |
| centi | c | \(10^{-2}\) |
| milli | m | \(10^{-3}\) |
| micro | \(\mu\) | \(10^{-6}\) |
| nano | n | \(10^{-9}\) |
How to convert a measurement
- Identify the kind of quantity: length, area, volume, capacity or mass.
- Find the factor between the two units. For area square the length factor; for volume cube it. Chain steps for distant units (km \(\to\) m \(\to\) cm).
- Decide: to a smaller unit multiply, to a larger unit divide.
- Check the size of the answer: smaller units give a bigger number.
Problems with mixed units
- Convert first: put every length in the same unit before you multiply or divide.
- Volume to capacity: work in \(\text{cm}^3\) to get mL, or in \(\text{m}^3\) to get kL (or \(\times 1000\) for L).
- Rates: divide by a rate such as litres per minute, then change the decimal part of a minute to seconds (\(\times 60\)).
- Per hectare or per kilogram: find the area in hectares or the mass in kilograms, multiply by the rate, then convert the answer (for example kg to t).
- Density: mass \(=\) density \(\times\) volume, with the volume in the unit the density uses (such as g per \(\text{cm}^3\)).
(i) \(0.36~\text{m}^2\) to \(\text{cm}^2\)
(ii) \(2500~\text{mm}^2\) to \(\text{cm}^2\)
(iii) \(0.0075~\text{m}^3\) to litres.
(i) The length factor is \(100\), so the area factor is \(100^2\). Smaller unit: multiply.
| \(0.36 \times 10\,000\) | \(=\) | \(\mathbf{3600}~\text{cm}^2\) |
(ii) \(1~\text{cm} = 10~\text{mm}\), so \(1~\text{cm}^2 = 10^2 = 100~\text{mm}^2\). Larger unit: divide.
| \(2500 \div 100\) | \(=\) | \(\mathbf{25}~\text{cm}^2\) |
(iii) Cube the factor, then use \(1~\text{cm}^3 = 1~\text{mL}\).
| \(0.0075 \times 100^3\) | \(=\) | \(7500~\text{cm}^3\) |
| \(\) | \(=\) | \(7500~\text{mL}\) |
| \(\) | \(=\) | \(\mathbf{7.5}~\text{L}\) |
(i) Write its width in millimetres, in scientific notation.
(ii) How many of these particles, side by side, would stretch across \(0.6\) mm?
(i) A millimetre is the larger unit, so divide by \(10^6\).
| \(120 \div 10^6\) | \(=\) | \(0.000\,12~\text{mm}\) |
| \(\) | \(=\) | \(\mathbf{1.2 \times 10^{-4}}~\text{mm}\) |
(ii) Put both lengths in the same unit before dividing.
| \(0.6~\text{mm}\) | \(=\) | \(0.6 \times 10^6 = 600\,000~\text{nm}\) |
| \(600\,000 \div 120\) | \(=\) | \(\mathbf{5000}\) |
Check with (i): \(0.6 \div (1.2 \times 10^{-4}) = 5000\) \(\checkmark\)
(i) Find its capacity in litres.
(ii) A hose fills it at \(9\) litres per minute. How long does it take to fill, in minutes and seconds?
(iii) What is the mass of the water?
(i) Change every length to metres first, then use \(1~\text{m}^3 = 1000~\text{L}\).
| \(V\) | \(=\) | \(0.9 \times 0.45 \times 0.5\) |
| \(\) | \(=\) | \(0.2025~\text{m}^3\) |
| \(\) | \(=\) | \(\mathbf{202.5}~\text{L}\) |
(ii) Divide by the rate and convert the decimal part of a minute.
| \(202.5 \div 9\) | \(=\) | \(22.5~\text{min}\) |
| \(0.5 \times 60\) | \(=\) | \(30~\text{s}\) |
It takes 22 min 30 s.
(iii) \(1\) L of water has a mass of \(1\) kg, so the water has a mass of 202.5 kg.
The rain forms a thin layer on the roof: a rectangular prism \(12\) mm deep. Change \(12\) mm to metres.
| \(12~\text{mm}\) | \(=\) | \(0.012~\text{m}\) |
| \(V\) | \(=\) | \(15 \times 8 \times 0.012\) |
| \(\) | \(=\) | \(1.44~\text{m}^3\) |
| \(\) | \(=\) | \(\mathbf{1440}~\text{L}\) |
\(1~\text{m}^3\) of water is \(1\) kL with a mass of \(1\) t, so the mass is 1.44 t.
Common pitfalls
Frequently asked questions
How many square centimetres are in a square metre?
There are \(10\,000~\text{cm}^2\) in \(1~\text{m}^2\), because a square metre is \(100\) cm by \(100\) cm and \(100 \times 100 = 10\,000\).
How many litres are in a cubic metre?
One cubic metre holds \(1000\) litres, which is one kilolitre. A cubic metre is \(1\,000\,000~\text{cm}^3\) and each \(\text{cm}^3\) holds \(1\) mL.
Do you multiply or divide when converting units?
Multiply when changing to a smaller unit, because you need more of them. Divide when changing to a larger unit.
How many square metres are in a hectare?
A hectare is \(10\,000~\text{m}^2\), the area of a square \(100\) m by \(100\) m. There are \(100\) hectares in a square kilometre.
What is a micrometre?
A micrometre is one millionth of a metre, or one thousandth of a millimetre. Its symbol is \(\mu\text{m}\). It is used for things like cells and fibres.
What is the mass of one litre of water?
One litre of water has a mass of about \(1\) kilogram, because each millilitre has a mass of \(1\) gram. So a cubic metre of water has a mass of about \(1\) tonne.