Metric Units
Metric units are built on prefixes: kilo means a thousand, centi a hundredth and milli a thousandth. This NSW Year 9 Core maths subtopic uses them under NESA outcome MA5‑MAG‑C‑01 and revises the square and cubic conversions from Stage 4, the part most students find hardest.
You will convert lengths, masses, volumes and capacities along the metric ladder, from tonnes to grams and kilometres to millimetres, and even to nanometres. You will see why \(1~\text{m}^2 = 10\,000~\text{cm}^2\) and \(1~\text{m}^3 = 1\,000\,000~\text{cm}^3\), because the factor is squared or cubed, and link \(\text{cm}^3\) and \(\text{m}^3\) to millilitres and litres. Problems include a kitchen floor's area in square centimetres and how many litres a swimming pool holds. It is part of Year 9 Core Surface Area and Volume.
Theory
Metric units are built from one base unit and a prefix that tells you the factor. This Year 9 maths page shows how to convert length, mass and capacity, why square and cubic units use the factor squared or cubed, and how to turn a volume into litres.
The metric system uses a base unit (metre, gram, litre) and a prefix in front of it.
kilo means a thousand, centi a hundredth, milli a thousandth and nano a thousand-millionth. So a kilometre is \(1000\) m and a millimetre is \(\dfrac{1}{1000}\) of a metre.
To change to a smaller unit, multiply, because it takes more of them. To change to a bigger unit, divide.
Area is measured in square units such as \(\text{cm}^2\), and volume in cubic units such as \(\text{m}^3\). Capacity is how much a container holds, in litres. One cubic centimetre holds exactly one millilitre.
| Measure | Conversions |
|---|---|
| Length | \(10\) mm \(= 1\) cm, \(100\) cm \(= 1\) m, \(1000\) m \(= 1\) km, \(1\) m \(= 10^9\) nm |
| Mass | \(1000\) mg \(= 1\) g, \(1000\) g \(= 1\) kg, \(1000\) kg \(= 1\) t |
| Capacity | \(1000\) mL \(= 1\) L, \(1000\) L \(= 1\) kL |
Square units square the length factor. Cubic units cube it:
Volume and capacity:
| Change | Length | Area | Volume |
|---|---|---|---|
| m to cm | \(\times 100\) | \(\times 10\,000\) | \(\times 1\,000\,000\) |
| cm to mm | \(\times 10\) | \(\times 100\) | \(\times 1000\) |
| km to m | \(\times 1000\) | \(\times 1\,000\,000\) | \(\times 10^9\) |
How to convert a metric unit
- Decide whether the new unit is smaller or bigger. Smaller: multiply. Bigger: divide.
- Find the length factor from the prefixes. For two or more steps, multiply the factors together.
- Square the factor for square units, or cube it for cubic units.
- Convert, then check the size: a smaller unit must give a bigger number.
(i) \(4.7\) km to metres
(ii) \(350\) mm to centimetres
(iii) \(2.6\) km to centimetres
(i) Metres are smaller than kilometres, so multiply.
| \(4.7 \times 1000\) | \(=\) | \(\mathbf{4700} \text{ m}\) |
(ii) Centimetres are bigger than millimetres, so divide.
| \(350 \div 10\) | \(=\) | \(\mathbf{35} \text{ cm}\) |
(iii) Two steps down the ladder: km to m, then m to cm.
| \(2.6 \text{ km}\) | \(=\) | \(2.6 \times 1000 \text{ m} = 2600 \text{ m}\) |
| \(2600 \text{ m}\) | \(=\) | \(2600 \times 100 \text{ cm}\) |
| \(\) | \(=\) | \(\mathbf{260\,000} \text{ cm}\) |
(i) \(6500\) mg to grams
(ii) \(3.2\) kL to millilitres
(iii) \(0.000\,003\) m to nanometres
(i) Grams are bigger than milligrams, so divide by \(1000\).
| \(6500 \div 1000\) | \(=\) | \(\mathbf{6.5} \text{ g}\) |
(ii) Two steps: kL to L, then L to mL. Each step is \(\times 1000\).
| \(3.2 \times 1000 \times 1000\) | \(=\) | \(\mathbf{3\,200\,000} \text{ mL}\) |
(iii) A nanometre is a thousand-millionth of a metre, so \(1 \text{ m} = 10^9 \text{ nm}\).
| \(0.000\,003 \times 10^9\) | \(=\) | \(\mathbf{3000} \text{ nm}\) |
(i) \(7 \text{ m}^2\) to \(\text{cm}^2\)
(ii) \(85\,000 \text{ cm}^2\) to \(\text{m}^2\)
The length factor is \(100\). For area, square it.
| \(1 \text{ m}^2\) | \(=\) | \(100^2 \text{ cm}^2 = 10\,000 \text{ cm}^2\) |
(i) To the smaller unit: multiply.
| \(7 \times 10\,000\) | \(=\) | \(\mathbf{70\,000} \text{ cm}^2\) |
(ii) To the bigger unit: divide.
| \(85\,000 \div 10\,000\) | \(=\) | \(\mathbf{8.5} \text{ m}^2\) |
Find the volume in cubic metres first.
| \(V\) | \(=\) | \(1.5 \times 0.8 \times 0.5\) |
| \(\) | \(=\) | \(0.6 \text{ m}^3\) |
Use the capacity link \(1 \text{ m}^3 = 1000 \text{ L}\).
| \(0.6 \times 1000\) | \(=\) | \(\mathbf{600} \text{ L}\) |
Check through cubic centimetres: \(0.6 \text{ m}^3 = 600\,000 \text{ cm}^3 = 600\,000\) mL \(= 600\) L.
Common pitfalls
Frequently asked questions
How many square centimetres are in a square metre?
There are \(10\,000\) square centimetres in a square metre, because a square metre is \(100\) cm by \(100\) cm.
How many litres are in a cubic metre?
One cubic metre holds \(1000\) litres. It is \(1\,000\,000 \text{ cm}^3\), each \(1 \text{ cm}^3\) is \(1\) mL, and \(1000\) mL is \(1\) L.
Do you multiply or divide to convert units?
Multiply when changing to a smaller unit and divide when changing to a bigger unit. km to m multiplies by \(1000\); mm to cm divides by \(10\).
What does the prefix milli mean?
Milli means one thousandth. A millimetre is \(\dfrac{1}{1000}\) of a metre, and the same goes for milligrams and millilitres.
How do you convert cubic centimetres to cubic millimetres?
\(1\) cm is \(10\) mm, so \(1 \text{ cm}^3 = 10^3 = 1000 \text{ mm}^3\). Multiply by \(1000\).