Density, Mass & Volume –

Density, mass and volume questions appear on both Foundation and Higher GCSE papers. Density is a compound measure that connects how heavy something is to how much space it takes up. The formula triangle works identically to speed-distance-time, so once you master one, the other follows naturally. This guide covers the formulas, unit conversions, and real-world contexts you will meet in the exam.

What Is Density?

Density measures how much mass is packed into a given volume. A material with high density, such as lead, has a lot of mass in a small volume. A material with low density, such as cork, has relatively little mass for its size.

The standard unit of density in GCSE Maths is grams per cubic centimetre (g/cm³) or kilograms per cubic metre (kg/m³). The formula linking the three quantities is straightforward, and you can use a formula triangle to remember the rearrangements.

Density problems frequently appear alongside volume questions. You may need to calculate the volume of a 3D shape first, then use the density formula to find the mass. Matching units before substituting is essential — if the volume is in cm³, the density must be in g/cm³ (not kg/m³).

Key Formulas

Step-by-Step Method

1. Identify which quantity you need to find (density, mass, or volume).
2. Check units — ensure mass and volume units are compatible with the density unit.
3. Substitute into the correct rearrangement of the formula.
4. Calculate and include the correct unit in your answer.

Worked Example 1 — Foundation Level

Question: A block of metal has a mass of 540 g and a volume of 200 cm³. Calculate the density.

Working: Density = Mass ÷ Volume Density = 540 ÷ 200 = 2.7 g/cm³.

Answer: The density of the metal is 2.7 g/cm³.

Worked Example 2 — Higher Level

Question: A gold bar has a density of 19.3 g/cm³ and a volume of 51.8 cm³. Find the mass of the gold bar in kilograms.

Working: Mass = Density × Volume = 19.3 × 51.8 = 999.74 g. Convert to kg: 999.74 ÷ 1000 = 0.99974 kg.

Answer: The mass of the gold bar is approximately 1.00 kg (to 3 s.f.).

Worked Example 3 — Exam Style

Question: A cylindrical container has radius 5 cm and height 12 cm. It is completely filled with oil of density 0.8 g/cm³. Find the mass of the oil. Give your answer to 3 significant figures.

Working: Volume of cylinder = πr²h = π × 5² × 12 = 300π = 942.478... cm³. Mass = Density × Volume = 0.8 × 942.478... = 753.982... g.

Answer: The mass of the oil is 754 g (to 3 s.f.).

Common Mistakes

• Mixing up units. If density is in kg/m³ and volume is in cm³, you must convert one before calculating. 1 m³ = 1,000,000 cm³ and 1 kg = 1,000 g.
• Forgetting to calculate volume first. Many exam questions give dimensions of a shape rather than the volume directly. Work out the volume before using D = M/V.
• Inverting the formula. Density = Mass ÷ Volume, not Volume ÷ Mass. Use the formula triangle: M on top, D and V on the bottom.
• Rounding volume too early. When the volume involves π, keep the exact value until the final step to avoid rounding errors.

Exam Tips

• Draw the formula triangle (M at the top, D × V at the bottom) in your working. It helps you rearrange correctly and earns method marks.
• If the question involves a composite shape, find the total volume by adding or subtracting component volumes before applying the density formula.
• Always state the unit of your final answer — density questions carry a unit mark.
• Remember the key conversion: 1 g/cm³ = 1000 kg/m³. This comes up when questions switch between metric units.
• On multi-step questions, calculate the volume first and write it down before moving to the density calculation.

Practice Questions

Q1 (Foundation): A stone has a mass of 300 g and a volume of 120 cm³. Find its density.

Q2 (Foundation): A liquid has a density of 1.2 g/cm³. A container holds 500 cm³ of the liquid. Find the mass.

Q3 (Higher): A cube of aluminium has side length 8 cm. Aluminium has a density of 2.7 g/cm³. Find the mass of the cube in kilograms.

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Related Topics

• Compound Measures Density and Pressure
• Speed Distance and Time
• Volume of 3D Shapes

Summary

• Density = Mass ÷ Volume. Rearrange using the formula triangle for mass or volume.
• Common units are g/cm³ and kg/m³. Always check that your units match before calculating.
• 1 g/cm³ = 1000 kg/m³. Convert units before substituting if they do not match.
• Many exam questions require you to calculate volume from given dimensions before using the density formula.
• Density is a compound measure — it combines mass and volume into a single rate.
• Draw the formula triangle and state the formula you are using to earn method marks.
• Include the correct unit in your final answer — density questions almost always carry a unit mark.
• Higher-tier questions often combine density with volume of cylinders, prisms or spheres — revise both topics together.