Units: Converting Between Metric and Imperial — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers converting between units of length, mass, capacity, area and volume, and between metric and imperial measures. By the end of this guide you should be able to convert confidently within the metric system and decide whether to multiply or divide.

You should also be able to use the standard metric-imperial approximations, convert units of area and volume correctly, handle compound units such as speed and density, and choose sensible units for a given quantity.

The organising idea is that the direction of a conversion is decided by the size of the units, not by memory. Going to a smaller unit means you need more of them, so the number gets bigger and you multiply. Going to a larger unit means fewer of them, so the number gets smaller and you divide. Checking the answer against that one principle catches nearly every error in the topic — if you convert metres to centimetres and the number goes down, something has gone wrong, whatever the arithmetic said.

Key terms and definitions

Metric — the decimal system of units: millimetres, centimetres, metres, kilometres, grams, kilograms, millilitres, litres.

Imperial — the older system: inches, feet, miles, pounds, ounces, pints, gallons.

Conversion factor — the number relating two units, such as 100 for metres to centimetres.

Compound unit — a unit built from two others, such as km/h or g/cm³.

Capacity — the amount a container holds, measured in millilitres and litres.

Density — mass divided by volume.

Core concepts

Metric conversions

Metric units convert by powers of ten, which is what makes the system convenient.

For length: 10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km.

For mass: 1000 mg = 1 g, 1000 g = 1 kg, 1000 kg = 1 tonne.

For capacity: 1000 ml = 1 litre, and usefully 1 ml occupies 1 cm³.

To convert, multiply going to a smaller unit and divide going to a larger one.

So 3.5 m = 350 cm, because centimetres are smaller and more are needed. And 2500 g = 2.5 kg, because kilograms are larger and fewer are needed.

Deciding the direction

Rather than memorising which way each conversion goes, ask whether the new unit is bigger or smaller than the old one.

Converting 4 km into metres: metres are much smaller, so the number must be much bigger — 4000.

Converting 750 mm into metres: metres are much bigger, so the number must be much smaller — 0.75.

That check takes a second and is more reliable than recalling a rule, particularly under exam pressure.

Area conversions are squared

This is where most marks are lost in the topic.

There are 100 cm in a metre, so there are 100 × 100 = 10,000 cm² in a square metre — not 100.

The reason is that an area is two lengths multiplied, so the conversion factor applies twice.

Likewise 1 cm² = 100 mm², since there are 10 mm in a centimetre and 10 × 10 = 100.

And 1 km² = 1,000,000 m², since 1000 × 1000 = 1,000,000.

Volume conversions are cubed

The same logic extends: a volume is three lengths multiplied, so the factor applies three times.

1 m³ = 1,000,000 cm³, since 100 × 100 × 100 = 1,000,000.

And 1 cm³ = 1000 mm³.

A useful bridge between volume and capacity: 1 litre = 1000 cm³, and 1 ml = 1 cm³ exactly. That equivalence turns many capacity problems into volume problems and back.

Metric and imperial

The specification expects a few approximations to be known, and questions usually supply anything unusual.

For length: 1 inch ≈ 2.5 cm, 1 foot ≈ 30 cm, 1 mile ≈ 1.6 km.

For mass: 1 kg ≈ 2.2 pounds.

For capacity: 1 litre ≈ 1.75 pints, and 1 gallon ≈ 4.5 litres.

Within the imperial system: 12 inches = 1 foot, 3 feet = 1 yard, 16 ounces = 1 pound, and 8 pints = 1 gallon.

These are approximate, so answers derived from them should not be quoted to excessive precision, and the ≈ symbol is appropriate.

The 5 miles ≈ 8 km relationship is worth knowing as a ratio, since it converts both ways without decimals: multiply by 8 and divide by 5 to go from miles to kilometres.

Compound units

A compound unit is built from two others, and converting one means converting both parts.

Speed in km/h converts to m/s by changing kilometres to metres and hours to seconds: multiply by 1000 and divide by 3600, which together means dividing by 3.6.

So 72 km/h is 72 ÷ 3.6 = 20 m/s.

Density is mass divided by volume, usually in g/cm³ or kg/m³. Converting between them requires both the mass and the volume factors, which is why the numbers change so dramatically: 1 g/cm³ is 1000 kg/m³.

Pressure and rates such as litres per minute work the same way.

When converting a compound unit, deal with the top and the bottom separately and combine at the end.

Density and mass problems

Density questions are the commonest place compound units are tested, and they combine conversion with a formula.

Density = mass ÷ volume, rearranged as mass = density × volume, and volume = mass ÷ density.

The units must agree before substituting. A block of volume 250 cm³ and density 7.8 g/cm³ has mass 250 × 7.8 = 1950 g, which is 1.95 kg.

Where the density is given in kg/m³ but the volume in cm³, one of them has to be converted first — and that conversion is a cubed one, so the factor is 1,000,000 rather than 100.

Stating the units at each stage is what keeps these straight, since the numbers alone give no clue that something has been missed.

Best-value and rate comparisons

Comparing prices or rates requires a common unit, and the choice of which unit to use is part of the method.

To compare a 750 ml bottle at £2.10 with a 2-litre bottle at £5.20, convert to the same unit first: 750 ml and 2000 ml.

Then find a rate both share — price per millilitre, or millilitres per pound. Per millilitre, the small bottle costs 0.28p and the large 0.26p, so the large bottle is better value.

Either rate works, provided you say which you used and compare consistently. Quoting the answer with its unit, and stating which item is better value and why, is what earns the final mark.

Choosing sensible units

Questions sometimes ask which unit is appropriate, and the answer is the one giving a manageable number.

The mass of a person is in kilograms rather than milligrams; the distance between cities is in kilometres rather than centimetres; the capacity of a teaspoon is in millilitres rather than litres.

An answer of 0.0000054 km or 5,400,000 mm signals that the wrong unit was chosen, even when the value is correct.

Converting before calculating

Where a problem mixes units, convert everything to a common unit first, before doing any arithmetic.

A rectangle measuring 80 cm by 1.2 m has an area found by converting the 1.2 m to 120 cm, giving 80 × 120 = 9600 cm². Multiplying 80 by 1.2 directly gives 96, which is meaningless.

Choosing which unit to convert to is worth a moment's thought — usually the one that avoids decimals or keeps the numbers smallest.

Worked examples

Example 1: An area conversion

A room has an area of 18 m². What is this in cm²?

Converting metres to centimetres multiplies by 100, so converting square metres to square centimetres multiplies by 100 × 100 = 10,000.

18 × 10,000 = 180,000 cm².

Using a factor of 100 would have given 1800 cm², which is a hundred times too small — and a quick sanity check exposes it, since a square metre alone is already 10,000 cm².

Example 2: A compound unit

A cheetah runs at 108 km/h. What is this in m/s?

Convert the kilometres to metres: 108 km is 108,000 m.

Convert the hours to seconds: 1 hour is 3600 seconds.

So the speed is 108,000 ÷ 3600 = 30 m/s.

The shortcut is to divide by 3.6, which combines the two steps: 108 ÷ 3.6 = 30. ✓

Example 3: Mixed units in one problem

A bottle holds 1.5 litres. A glass holds 240 cm³. How many full glasses can be poured from the bottle?

Convert to a common unit. Since 1 litre = 1000 cm³, the bottle holds 1500 cm³.

Divide: 1500 ÷ 240 = 6.25.

Only whole glasses can be filled, so the answer is 6 full glasses, with 60 cm³ left over.

The rounding here is downward because a partly filled glass does not count as a full one — the context decides, not the usual rounding rule.

Common mistakes and how to avoid them

Multiplying when you should divide. Ask whether the new unit is bigger or smaller, and check the answer's size.

Using the length factor for an area. Square it: 1 m² = 10,000 cm².

Using the length factor for a volume. Cube it: 1 m³ = 1,000,000 cm³.

Converting only one part of a compound unit. Both the top and the bottom must change.

Calculating before converting. Bring everything to a common unit first.

Quoting an imperial conversion too precisely. These factors are approximate.

Choosing an absurd unit. If the number needs many zeros or many decimal places, change the unit.

Exam technique for "Units and Conversion"

Write the conversion factor down as a separate step before using it, and state whether you are multiplying or dividing.

For area and volume, write the factor as a square or a cube — "100² = 10,000" — so the reasoning is visible and the mark is secured.

Sense-check every answer against the size of the units: smaller unit, bigger number.

When a problem mixes units, convert everything first and note which unit you chose.

For compound units, handle the numerator and denominator separately, then combine.

Use ≈ for imperial conversions and round the answer sensibly rather than quoting every digit.

Quick revision summary

The direction is decided by the size of the units: to a smaller unit, multiply (more of them); to a larger unit, divide (fewer of them).

Metric length: 10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km. Mass: 1000 g = 1 kg. Capacity: 1000 ml = 1 litre.

Area conversions are squared and volume conversions are cubed: 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³.

Useful bridge: 1 ml = 1 cm³, so 1 litre = 1000 cm³.

Imperial approximations: 1 inch ≈ 2.5 cm, 1 foot ≈ 30 cm, 5 miles ≈ 8 km, 1 kg ≈ 2.2 lb, 1 litre ≈ 1.75 pints.

For compound units, convert both parts: km/h to m/s means dividing by 3.6.

Convert to a common unit before calculating, and choose units that keep the number manageable.