Density, explained with examples
Density tells you how much mass is packed into a given volume. It explains why a steel ship floats while a steel bolt sinks, and it is one of the most common GCSE Physics calculations.
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The density equation
density = mass / volume, or ρ = m/V
The symbol ρ is the Greek letter rho. Mass is measured in kilograms (kg) and volume in cubic metres (m³), giving density in kg/m³. In practical work, grams and cubic centimetres are often more convenient, giving density in g/cm³. For GCSE Physics and Combined Science exams in 2025, 2026 and 2027, an equations sheet is provided in the exam, including the equations students previously had to recall; check the arrangements for your exam year, but learn it anyway, because fluent recall saves time.
Rearranged, it gives m = ρV and V = m/ρ.
Converting units
1 g/cm³ = 1000 kg/m³. To convert from g/cm³ to kg/m³, multiply by 1000. Water has a density of 1 g/cm³, which is 1000 kg/m³. The reason: 1 m³ is 100 × 100 × 100 = 1,000,000 cm³, and 1 kg is 1000 g, so the factor is 1,000,000 ÷ 1000 = 1000.
Some typical densities: air about 1.2 kg/m³, wood about 600 kg/m³, water 1000 kg/m³, aluminium 2700 kg/m³, iron 7900 kg/m³ and gold 19,300 kg/m³.
Worked examples
Example 1: finding density
A block has a mass of 540 g and a volume of 200 cm³. Find its density in g/cm³ and kg/m³.
- ρ = 540/200 = 2.7 g/cm³.
- In kg/m³: 2.7 × 1000 = 2700 kg/m³. This matches aluminium.
Example 2: finding mass
A tank holds 0.5 m³ of water. What is the mass of the water?
- m = ρV = 1000 × 0.5 = 500 kg.
Example 3: finding volume
A gold bar has a mass of 12.5 kg. Gold has density 19,300 kg/m³. Find its volume.
- V = m/ρ = 12.5/19,300 ≈ 6.5 × 10−4 m³, or about 650 cm³.
Example 4: a regular shape
A cuboid of metal measures 2 cm × 3 cm × 5 cm and has a mass of 237 g. Find its density and suggest the metal.
- Volume = 2 × 3 × 5 = 30 cm³.
- ρ = 237/30 = 7.9 g/cm³, which suggests iron.
Example 6: a two-step problem
A solid copper cylinder has radius 2 cm and height 10 cm. Copper has density 8.96 g/cm³. Find its mass in kilograms.
- Volume = πr²h = π × 4 × 10 ≈ 125.7 cm³.
- m = ρV = 8.96 × 125.7 ≈ 1126 g = 1.13 kg. Many exam questions combine density with an area or volume formula from maths.
Comparing materials
Density is a property of a material, not of a particular object. A small iron nail and a large iron bar have the same density, even though their masses are very different. That is why density can help identify an unknown material: calculate it, then compare with a table of known values.
Density and the particle model
The particle model explains why solids, liquids and gases have different densities. In a solid, particles are closely packed in a regular arrangement, so a lot of mass fits into a small volume. In a liquid, particles are still close together but can move past each other, so liquids are usually slightly less dense than their solids. In a gas, particles are far apart and move quickly and randomly, so gases have very low densities, often about a thousand times less than the solid or liquid.
Water is an important exception: ice is less dense than liquid water, because the molecules form an open structure when they freeze. That is why ice floats, and why lakes freeze from the top down.
When a substance changes state, its mass is conserved. The number of particles stays the same; only their arrangement and spacing change. So melting or boiling changes the volume, and therefore the density, but not the mass.
The density required practical
Regular objects: measure the length, width and height with a ruler or vernier callipers, calculate the volume, measure the mass on a balance, then use ρ = m/V.
Irregular objects: use a displacement (eureka) can. Fill the can with water up to the spout, lower the object in gently, and collect the water that overflows in a measuring cylinder. The volume of water displaced equals the volume of the object. Measure the mass on a balance, then calculate the density.
Liquids: place a measuring cylinder on a balance and zero (tare) it. Pour in a known volume of liquid, read the volume at eye level from the bottom of the meniscus, and read the mass from the balance.
Example 5: displacement
A stone has a mass of 156 g. When placed in a displacement can, it pushes out 60 cm³ of water. Find its density.
- ρ = 156/60 = 2.6 g/cm³.
Improving accuracy
- Read the measuring cylinder at eye level to avoid parallax errors.
- Use a measuring cylinder with a small scale division for small volumes.
- Make sure no water is lost by splashing when the object goes in.
- Repeat measurements and calculate a mean.
Floating and sinking
An object floats in a liquid if its average density is less than the density of the liquid, and sinks if it is greater. A steel ship floats because its hull contains a large volume of air, making its average density less than that of water. Hot air balloons rise because hot air is less dense than the cooler air around it.
Common mistakes
- Mixing units, such as grams with cubic metres.
- Multiplying by 1,000,000 instead of 1000 when converting g/cm³ to kg/m³.
- Reading the meniscus from the top instead of the bottom.
- Saying mass changes during a change of state; only volume and density change.
Practice questions
Try each one before opening the answer.
Q1. Find the density of an object with mass 2.4 kg and volume 0.003 m³.
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Q2. Convert 13.6 g/cm³ (mercury) to kg/m³.
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Q3. A 50 cm³ sample of oil has a mass of 46 g. Will it float on water?
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Q4. Find the mass of 2 m³ of air (density 1.2 kg/m³).
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Q5. Why is a gas much less dense than a solid?
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Frequently asked questions
Is density in Combined Science and separate Physics?
Yes. Density and the density required practical are in the particle model of matter topic for both.
Is the density equation on the GCSE equation sheet?
For GCSE exams in 2025, 2026 and 2027, an equations sheet is provided with the Physics and Combined Science papers, including the equations students previously had to recall. Check the arrangements for your exam year, and learn the equation anyway, because using it fluently saves time.
Key takeaways
- ρ = m/V, in kg/m³ or g/cm³.
- 1 g/cm³ = 1000 kg/m³.
- Use a displacement can for irregular objects.
- Particle spacing explains why gases are much less dense.
- Objects float if their average density is less than the liquid's.