Compound measures combine two or more different units into a single measurement — and in GCSE Maths, the two you need to know inside out are density and pressure. These questions appear on both Foundation and Higher papers across AQA, Edexcel and OCR, and they reward students who can rearrange formulas confidently and keep units consistent. This guide breaks down both formulas, shows you the triangle method for rearranging, provides worked examples at each tier, and gives you practice questions to solidify your understanding. If speed, distance and time is the other compound measure you need, we cover that in a separate topic guide. For an overview of key formulas, see our GCSE Maths formulas guide.
What Are Compound Measures?
A compound measure is a measurement that involves more than one unit. Speed (miles per hour), density (grams per cubic centimetre), and pressure (newtons per square metre) are all compound measures.
Density
Density is the mass per unit volume of a substance.
$$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$$
The most common units at GCSE are g/cm³ (grams per cubic centimetre) or kg/m³ (kilograms per cubic metre).
From this formula you can also find:
$$\text{Mass} = \text{Density} \times \text{Volume}$$
$$\text{Volume} = \frac{\text{Mass}}{\text{Density}}$$
Pressure
Pressure is the force per unit area.
$$\text{Pressure} = \frac{\text{Force}}{\text{Area}}$$
Common units are N/m² (newtons per square metre, also called pascals) or N/cm².
Rearranging gives:
$$\text{Force} = \text{Pressure} \times \text{Area}$$
$$\text{Area} = \frac{\text{Force}}{\text{Pressure}}$$
The Triangle Method
Both density and pressure have the same structure as Speed = Distance ÷ Time. Place the quantity that is divided (Mass or Force) at the top of the triangle, and the other two at the bottom. Cover what you need to find, and the remaining layout shows your formula.
Step-by-Step Method
Density Problems
- Identify the values given and which quantity you need to find.
- Choose the correct formula arrangement (D = M/V, M = D × V, or V = M/D).
- Check units — if mass is in kg and volume in cm³, you may need to convert.
- Substitute and calculate.
- Write the answer with correct compound units.
Pressure Problems
- Identify force, area, and pressure — decide which is unknown.
- Select the correct formula (P = F/A, F = P × A, or A = F/P).
- Check units are consistent.
- Substitute and calculate.
- Include the correct compound units in your answer.
Worked Example 1 — Foundation Level
A metal block has a mass of 540 g and a volume of 200 cm³. Calculate the density of the metal.
Step 1: Write the formula: Density = Mass ÷ Volume.
Step 2: Substitute: Density = 540 ÷ 200 = 2.7.
Step 3: Write the answer with units: 2.7 g/cm³.
Follow-up
A different block of the same metal has a volume of 350 cm³. What is its mass?
Mass = Density × Volume = 2.7 × 350 = 945 g.
Worked Example 2 — Higher Level
A solid cylinder has a radius of 5 cm and a height of 12 cm. It has a density of 8.9 g/cm³. Calculate the mass of the cylinder. Give your answer to 3 significant figures.
Step 1: Find the volume of the cylinder. $$V = \pi r^2 h = \pi \times 5^2 \times 12 = 300\pi = 942.478 \text{ cm}^3$$
Step 2: Use Mass = Density × Volume. $$M = 8.9 \times 942.478 = 8,388.05 \text{ g}$$
Step 3: Round to 3 significant figures: 8,390 g (or 8.39 kg).
Pressure Extension
The cylinder is placed on a table with its circular face down. Calculate the pressure it exerts on the table. Take g = 9.8 N/kg.
Step 1: Convert mass to force (weight): Force = mass × g = 8.39 × 9.8 = 82.222 N.
Step 2: Area of circular base: A = π × 5² = 78.54 cm².
Step 3: Pressure = Force ÷ Area = 82.222 ÷ 78.54 = 1.047 N/cm² (3 d.p.).
Common Mistakes
- Mixing up the formula. Students sometimes put volume on top or mass on the bottom. Remember: Density = Mass ÷ Volume — the "heavier" quantity (mass) is on top.
- Unit mismatches. If mass is in kilograms and volume in cm³, your density will not be in standard units. Convert before dividing.
- Forgetting compound units. Writing "2.7" without "g/cm³" loses the units mark.
- Confusing density and pressure formulas. They have the same structure but use different quantities. Read the question to determine which formula applies.
- Not calculating area correctly. In pressure questions, students sometimes use length instead of area. Pressure requires a two-dimensional measurement (cm², m²).
Exam Tips
- State the formula at the start. Even if you make a calculation error, the formula earns method marks.
- Set out conversions clearly. If you need to convert kg to g (×1000) or m² to cm² (×10,000), write it as a separate line.
- Combined questions on Higher papers may give you a 3D shape, ask you to find the volume, then use the volume to find mass or pressure. Work through each step methodically.
- Read the units in the question. Sometimes the question gives density in kg/m³ but lengths in cm — you will need to convert.
- Pressure questions may involve weight. Remember: Weight = mass × g (where g = 9.8 N/kg or 10 N/kg as stated in the question).
Practice Questions
Question 1 (Foundation) A stone has a mass of 420 g and a volume of 150 cm³. What is its density?
Question 2 (Foundation) A box exerts a force of 200 N on the ground. The base of the box is 0.5 m by 0.4 m. Calculate the pressure on the ground.
Question 3 (Higher) A gold bar has a mass of 12.44 kg and a density of 19.3 g/cm³. Calculate the volume of the bar in cm³. Give your answer to 1 decimal place.
Question 4 (Higher) A triangular prism has a cross-sectional area of 30 cm², length 20 cm, and is made of aluminium with density 2.7 g/cm³. It rests on its triangular face. Calculate the pressure it exerts on the surface. Take g = 9.8 N/kg.
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Related Topics
- Speed, Distance and Time — the third key compound measure at GCSE
- Volume of 3D Shapes — needed to calculate density in many questions
- Unit Conversions — essential for matching units across a problem
- Area of 2D Shapes — needed for pressure calculations
Summary
Compound measures combine two or more units. Density = Mass ÷ Volume and Pressure = Force ÷ Area are the two main compound measure formulas at GCSE besides speed. Use the triangle method to rearrange confidently, always check that units are consistent before substituting, and include compound units in your final answer. On Higher papers, expect these formulas to be combined with volume or area calculations for 3D shapes. Practise converting between kg and g, and between m² and cm², so these steps become automatic under exam conditions.