What you'll learn
This topic covers converting between units — metric and imperial — and working with compound measures such as speed, density and pressure. In this guide you will learn how to convert lengths, areas, volumes and other quantities, how to handle compound units, and how to use the formulae for speed, density and pressure. These practical skills appear throughout GCSE Maths and science.
Key terms and definitions
Metric units — units based on tens (mm, cm, m, km; g, kg; ml, l).
Imperial units — older units (inches, feet, miles; pounds, ounces; pints, gallons).
Compound measure — a measure made from two others (e.g. speed = distance/time).
Density — mass per unit volume (density = mass ÷ volume).
Pressure — force per unit area (pressure = force ÷ area).
Core concepts
Metric conversions
Metric units convert by powers of ten: 10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km; 1000 g = 1 kg; 1000 ml = 1 litre. Multiply when converting to smaller units and divide when converting to larger units. For example, 3.5 m = 350 cm; 2500 g = 2.5 kg.
Imperial and metric conversions
You may need approximate conversions such as 5 miles ≈ 8 km, 1 kg ≈ 2.2 pounds, and 1 litre ≈ 1.75 pints. Use the given conversion factor and multiply or divide as appropriate, keeping track of which unit is larger.
Converting areas and volumes
Area and volume conversions use squared and cubed factors. Since 1 m = 100 cm, 1 m² = 100² = 10 000 cm², and 1 m³ = 100³ = 1 000 000 cm³. A common error is using the linear factor (100) for area or volume — you must square or cube it.
Compound measures
A compound measure combines two quantities. The three most common are:
- Speed = distance ÷ time (units like m/s, km/h).
- Density = mass ÷ volume (units like g/cm³, kg/m³).
- Pressure = force ÷ area (units like N/m², called pascals).
Each can be rearranged: distance = speed × time; mass = density × volume; force = pressure × area.
Using the formula triangles
For each compound measure, the three quantities form a triangle (e.g. distance on top, speed and time below) so you can cover the one you want to see whether to multiply or divide. Always check the units match — convert first if needed (e.g. time in hours for km/h).
Worked examples
Example 1: Metric conversion
Convert 4.2 km to metres.
4.2 × 1000 = 4200 m.
Example 2: Speed
A car travels 150 km in 2 hours. Find its average speed.
Speed = distance ÷ time = 150 ÷ 2 = 75 km/h.
Example 3: Density
A block has mass 240 g and volume 30 cm³. Find its density.
Density = mass ÷ volume = 240 ÷ 30 = 8 g/cm³.
Common mistakes and how to avoid them
Multiplying instead of dividing (or vice versa). Check which unit is larger before converting.
Using linear factors for area/volume. 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³.
Mismatched units in compound measures. Convert so the units are consistent (e.g. time in hours for km/h).
Mixing up the formula. Use the formula triangle to decide multiply or divide.
Forgetting the units in the answer. Always state the correct compound unit.
Exam technique for Units and Compound Measures
Know the metric conversions (powers of ten) and any given imperial factors.
Square or cube the factor for area or volume conversions.
Use the right compound formula — speed, density or pressure.
Check and convert units before calculating.
State the units in your final answer.
Quick revision summary
Metric units convert by powers of ten (10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km, 1000 g = 1 kg, 1000 ml = 1 l): multiply for smaller units, divide for larger. Imperial–metric conversions use given factors (5 miles ≈ 8 km, 1 kg ≈ 2.2 lb). For areas and volumes, square or cube the linear factor — 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³ — a frequent source of error. Compound measures combine two quantities: speed = distance ÷ time, density = mass ÷ volume, pressure = force ÷ area, each rearrangeable via a formula triangle (cover the quantity you want). Always ensure the units are consistent (e.g. time in hours for km/h) before calculating, and state the correct compound unit in your answer. The common mistakes are converting the wrong way, using linear factors for area/volume, mismatched units, and forgetting units. Know the conversions, square/cube for area/volume, use the right formula, and keep units consistent.