Converting Units
Converting Units in NSW Year 10 Core maths is about moving confidently between metric units of length, area, volume and capacity (NESA outcome MA5‑MAG‑C‑01). It also introduces prefixes for huge and tiny units, such as mega, micro and nano, and writes the results in scientific notation where that helps.
You will convert lengths such as \(47\,500\) cm to km, change square and cubic units like \(\text{m}^2\) to \(\text{cm}^2\) and \(\text{m}^3\) to \(\text{cm}^3\), and work with hectares, megawatts, micrometres and nanometres. You will also solve practical problems: the thickness of one sheet of paper, the litres of rain collected from a shed roof, the cost of mulch for a garden bed and the water lost from a dripping tap. It is part of Year 10 Core Measurement.
Theory
To convert units, multiply when you change to a smaller unit and divide when you change to a bigger unit. This Year 10 maths page covers length, area and volume units, hectares, litres and cubic metres, and prefixes such as mega, micro and nano with answers in scientific notation.
The metric system builds every unit from a base unit (metre, gram, litre, second) and a prefix. The prefix is a power of ten: kilo means \(1000\), centi means one hundredth and milli means one thousandth.
To convert to a smaller unit, multiply: there will be more of them. To convert to a bigger unit, divide: there will be fewer.
Area units are squares, so the length factor is squared. Volume units are cubes, so it is cubed. Since \(1\text{ m} = 100\text{ cm}\), \(1\text{ m}^2 = 100^{2}\text{ cm}^2\) and \(1\text{ m}^3 = 100^{3}\text{ cm}^3\).
A hectare (ha) is the area of a square \(100\) m by \(100\) m, so \(1\text{ ha} = 10\,000\text{ m}^2\). Capacity measures how much a container holds: \(1\text{ mL} = 1\text{ cm}^3\) and \(1\text{ m}^3 = 1000\) L.
Very large and very small measurements, such as a wavelength in nanometres or power in megawatts, are often written in scientific notation, for example \(3.6 \times 10^{-7}\).
Metric prefixes:
| Prefix | Symbol | Means | Example |
|---|---|---|---|
| giga | G | \(10^{9}\) | \(1\text{ GB} = 10^{9}\) bytes |
| mega | M | \(10^{6}\) | \(1\text{ MW} = 10^{6}\) W |
| kilo | k | \(10^{3}\) | \(1\text{ km} = 1000\) m |
| centi | c | \(10^{-2}\) | \(1\text{ cm} = 0.01\) m |
| milli | m | \(10^{-3}\) | \(1\text{ mL} = 0.001\) L |
| micro | µ | \(10^{-6}\) | \(1\ \mu\text{g} = 10^{-6}\) g |
| nano | n | \(10^{-9}\) | \(1\text{ ns} = 10^{-9}\) s |
Conversion facts to know:
| Measure | Key facts |
|---|---|
| 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 | \(1\text{ cm}^3 = 1000\text{ mm}^3\), \(1\text{ m}^3 = 10^{6}\text{ cm}^3\) |
| Capacity | \(1\text{ L} = 1000\text{ mL}\), \(1\text{ mL} = 1\text{ cm}^3\), \(1\text{ m}^3 = 1000\text{ L}\) |
| Mass | \(1\text{ t} = 1000\text{ kg}\), \(1\text{ kg} = 1000\text{ g}\), \(1\text{ g} = 1000\text{ mg}\), \(1\text{ mg} = 1000\ \mu\text{g}\) |
| Time | \(1\text{ min} = 60\text{ s}\), \(1\text{ h} = 60\text{ min}\), \(1\text{ day} = 24\text{ h}\), \(1\text{ week} = 7\text{ days}\) |
How to convert a measurement
- Decide whether the new unit is bigger or smaller. Smaller unit: multiply. Bigger unit: divide.
- Find the factor. Step through one unit at a time (km to m to cm), and square or cube the length factor for area or volume.
- Calculate, and write very large or very small answers in scientific notation if asked.
- Check the size makes sense: more of a small unit, fewer of a big unit.
Measurement problems
- Same units first. Before you add, multiply or divide two measurements, write them in the same unit.
- How many fit? Divide the big length by the small length, both in the same unit.
- Rain or a layer on a flat surface is a thin rectangular prism: volume \(=\) length \(\times\) width \(\times\) depth, with the depth changed to metres. Then convert \(\text{m}^3\) to litres.
- Cost: find the volume in cubic metres, then multiply by the price per cubic metre: \(0.5\text{ m}^3\) of soil at $80 per cubic metre costs $40.
- A rate over time (drips per second, litres per minute): count the time units first, such as \(60 \times 60 \times 24\) seconds in a day, then multiply.
(i) \(0.86\) km to cm
(ii) \(4500\) mm to m
Move one unit at a time. Multiply going to a smaller unit, divide going to a bigger unit.
(i) km to m is \(\times 1000\), then m to cm is \(\times 100\).
| \(0.86 \times 1000\) | \(=\) | \(860\text{ m}\) |
| \(860 \times 100\) | \(=\) | \(\mathbf{86\,000}\text{ cm}\) |
(ii) mm to m is \(\div 1000\) (there are \(1000\) mm in a metre).
| \(4500 \div 1000\) | \(=\) | \(\mathbf{4.5}\text{ m}\) |
Sense check: centimetres are small, so there should be lots of them.
(i) \(2.5\text{ m}^2\) to \(\text{cm}^2\)
(ii) \(3400\text{ mm}^2\) to \(\text{cm}^2\)
(iii) \(0.75\text{ m}^3\) to \(\text{cm}^3\)
Square the length factor for area and cube it for volume.
(i) \(1\text{ m}^2 = 100 \times 100 = 10\,000\text{ cm}^2\)
| \(2.5 \times 10\,000\) | \(=\) | \(\mathbf{25\,000}\text{ cm}^2\) |
(ii) \(1\text{ cm}^2 = 10 \times 10 = 100\text{ mm}^2\), and mm² to cm² is a bigger unit, so divide.
| \(3400 \div 100\) | \(=\) | \(\mathbf{34}\text{ cm}^2\) |
(iii) \(1\text{ m}^3 = 100 \times 100 \times 100 = 10^{6}\text{ cm}^3\)
| \(0.75 \times 10^{6}\) | \(=\) | \(750\,000\) |
| \(\) | \(=\) | \(\mathbf{7.5 \times 10^{5}}\text{ cm}^3\) |
(i) in micrometres
(ii) in metres, in scientific notation. (\(1\ \mu\text{m} = 10^{-6}\) m)
There are \(1000\ \mu\text{m}\) in \(1\) mm, because \(10^{-3} \div 10^{-6} = 10^{3}\).
(i) mm to µm is a smaller unit, so multiply.
| \(0.07 \times 1000\) | \(=\) | \(\mathbf{70}\ \mu\text{m}\) |
(ii) mm to m is a bigger unit, so divide by \(1000\).
| \(0.07 \div 1000\) | \(=\) | \(0.000\,07\text{ m}\) |
| \(\) | \(=\) | \(\mathbf{7 \times 10^{-5}}\text{ m}\) |
Check: \(70 \times 10^{-6} = 7 \times 10^{-5}\), the same answer.
Put every length in metres first, find the volume in cubic metres, then change to litres.
| \(60\text{ cm}\) | \(=\) | \(0.6\text{ m}\) |
| \(V\) | \(=\) | \(2.4 \times 1.5 \times 0.6\) |
| \(\) | \(=\) | \(2.16\text{ m}^3\) |
| \(2.16 \times 1000\) | \(=\) | \(\mathbf{2160}\text{ L}\) |
Mixing \(60\) cm with metres would give \(216\text{ m}^3\), a pond the size of a house. Convert first.
Common pitfalls
Frequently asked questions
How do you convert units in maths?
Work out whether the new unit is bigger or smaller. To change to a smaller unit multiply by the conversion factor; to change to a bigger unit divide. Step through one unit at a time if needed.
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?
There are \(1000\) litres in \(1\text{ m}^3\). A litre is \(1000\text{ cm}^3\) and a cubic metre is \(10^{6}\text{ cm}^3\).
What is a hectare?
A hectare is \(10\,000\text{ m}^2\), the area of a square \(100\) m by \(100\) m. It is used for farms, parks and building blocks. To change square metres to hectares, divide by \(10\,000\).
What do micro and nano mean?
Micro (µ) means one millionth, \(10^{-6}\). Nano (n) means one billionth, \(10^{-9}\). So \(1\ \mu\text{m} = 10^{-6}\) m and \(1\text{ nm} = 10^{-9}\) m.
Do you multiply or divide to convert mm to m?
Divide. A metre is a bigger unit, so you get fewer of them. There are \(1000\) mm in a metre, so divide by \(1000\).