Density: revision notes and practice

What you will learn

• Explain density using mass, volume and a particle model.
• Calculate density, mass or volume and convert cubic units correctly.
• Plan density measurements for regular solids, irregular solids and liquids.

Key rules

Density equation

ρ = m/V. Mass in kg and volume in m³ give density in kg/m³. Mass in g and volume in cm³ give density in g/cm³.

Find mass or volume

Rearrange to mass m = ρV or volume V = m/ρ. Volume of a cuboid = length × width × height, using matching length units.

Convert cubic units

1 cm³ = 0.000001 m³ because 1 cm = 0.01 m and (0.01)³ = 0.000001. Also, 1 g/cm³ = 1000 kg/m³.

Worked examples

Example 1: A block has mass 270 g and volume 100 cm³. Find density in g/cm³, then kg/m³.

1. ρ = 270 ÷ 100 = 2.7 g/cm³.
2. Multiply by 1000 to convert density to kg/m³.

2.7 g/cm³ = 2700 kg/m³

Example 2: A liquid has density 800 kg/m³ and volume 0.003 m³. Find its mass.

1. Use m = ρV.
2. m = 800 × 0.003 = 2.4 kg.

2.4 kg

A practice question

An object has mass 108 kg and volume 12 m³. Calculate its density in kg/m³.

1. Density compares equal volumes

Density is mass per unit volume. Imagine identical 1 cm³ samples: the denser material has more mass in that same space. A heavy object is not automatically dense, because it may also have a large volume. A small metal sample can be denser than a much heavier foam block.

Density depends on the material and conditions, rather than simply the amount you have. Cutting a uniform block in half halves both its mass and volume, leaving their ratio unchanged. For a hollow or mixed object, mass divided by its outer volume gives an average density that includes spaces inside.

ρ = m/V; ρ is the Greek letter rho

2. Choose compatible mass and volume units

Mass in kilograms divided by volume in cubic metres gives density in kg/m³. Mass in grams divided by volume in cubic centimetres gives g/cm³. A 360 g block occupying 120 cm³ has density 3 g/cm³. Writing 3 kg/m³ would change the meaning by a factor of 1000.

Rearrange to m = ρV or V = m/ρ. A liquid with density 900 kg/m³ occupying 0.002 m³ has mass 1.8 kg. Before substituting, check that the volume unit matches the density denominator. Do not combine kg/m³ with cm³ without converting one of them.

ρ = m/V; m = ρV; V = m/ρ

3. Cubic conversions need a cubed factor

One centimetre is 0.01 m, so a cube measuring 1 cm on every side has volume (0.01)³ = 0.000001 m³. Therefore divide cm³ by one million to get m³. Convert grams to kilograms separately by dividing by 1000. A length conversion factor cannot be reused unchanged for volume.

These two conversions explain why 1 g/cm³ = 1000 kg/m³. Multiply a density in g/cm³ by 1000 to express it in kg/m³. Also, 1 mL = 1 cm³ and 1 litre = 0.001 m³. Write an intermediate conversion line when the question mixes litres, grams and SI units.

4. Measure a regular solid from its dimensions

Measure mass using a suitable balance, checking that it reads zero before the object is added. For a cuboid, measure length, width and height, then multiply them using consistent length units. A block of 4 cm × 3 cm × 2 cm occupies 24 cm³; at 72 g its density is 3 g/cm³.

Repeat dimensions at different positions if the object is not perfectly uniform. A ruler suits larger lengths; a micrometer or callipers may suit a small thickness or diameter when your course uses them. Avoid rounding each dimension early, because their uncertainties affect the calculated volume and density.

Cuboid volume = length × width × height

5. Use displacement for an irregular solid

For an irregular waterproof object, measure mass dry, then find its volume by the water it displaces. In a measuring cylinder, note the initial water volume, fully submerge the object and read the new volume. A rise from 40 mL to 58 mL means the object occupies 18 cm³.

Read the bottom of the water meniscus at eye level, avoid splashes and remove trapped bubbles. The method assumes the object does not dissolve or absorb water. An attached air bubble increases measured displacement and can make calculated density too low. Objects that float need an adapted method that accounts for any extra sinker.

6. Subtract the container when measuring liquids

Measure an empty container's mass, then the mass of that container with a known volume of liquid. Subtract to find the liquid's mass. If the empty cylinder is 80 g and the filled cylinder is 125 g, the liquid mass is 45 g. At a volume of 50 cm³, density is 0.9 g/cm³.

You can instead tare the balance with the empty container in place if the apparatus permits. Use the same container, clean its outside and keep the liquid level stable. Including the container's mass makes density too large. Use appropriate school procedures for any liquid; an unknown substance is not suitable for a home investigation.

7. Explain density with a particle model

In the GCSE model, solid particles vibrate around positions, liquid particles remain close but move past one another, and gas particles are much farther apart. Gases usually have much lower density because a given mass occupies much more space. Density differences also depend on particle masses, so spacing alone is not the whole explanation.

Heating often increases a material's volume while its mass stays constant, reducing density. There are exceptions: water expands when freezing, so ice is less dense than liquid water. During a change of state in a closed sample, mass is conserved even if volume and density change. Do not say the particles themselves expand or become lighter.

8. Check the result and its limitations

For the same mass, a larger volume gives a smaller density; for the same volume, a larger mass gives a larger density. Use these comparisons to check your answer. If your measured volume is too large, m/V is too small. A reading error has a greater percentage effect on a tiny volume than on a large one.

Repeat the measurement, calculate a mean where appropriate and explain why unusual values need investigation. Record mass and volume with units before deriving density. Floating depends on average density and buoyancy: an object less dense on average than the surrounding liquid can float. Detailed upthrust and fluid-pressure calculations belong to further forces revision.

Check your understanding

How to earn the marks

• Check whether the stated volume belongs to the material or includes a hollow space.
• Convert each length before calculating an SI volume, or convert the finished cubic volume correctly.
• Subtract initial water level or empty-container mass when the method requires it.
• Explain the direction of an error: too much measured volume makes calculated density too small.

A common mistake

Converting cm³ to m³ is not the same as converting cm to m. Cube the conversion factor.

Course context and official specifications

This guide covers selected skills. Some extensions are Higher content; choose your official tier separately from practice difficulty in the app. These are original questions and explanations.

• AQA GCSE Physics — 4.3.1.1

  The core calculations in this bank are selected separate Physics skills. Foundation and Higher content differ within these larger sections; this is not complete topic coverage.

• Edexcel GCSE Physics — 14.2–14.4

  The core calculations in this bank are selected separate Physics skills. Foundation and Higher content differ within these larger sections; this is not complete topic coverage.

• OCR Gateway GCSE Physics A — P1.1; PM1.1i

  The core calculations in this bank are selected separate Physics skills. Foundation and Higher content differ within these larger sections; this is not complete topic coverage.

• OCR Twenty First Century GCSE Physics B — P6.1.1–P6.1.2

  The core calculations in this bank are selected separate Physics skills. Foundation and Higher content differ within these larger sections; this is not complete topic coverage.

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