Two objects can be exactly the same size and still have wildly different masses. Density is the quantity that captures this — how much mass is squeezed into a given amount of space.
📘 What you need to know
Density is the mass per unit volume of an object
It’s calculated using ρ = m⁄V
Units depend on the units used for mass and volume — commonly g cm⁻³ or kg m⁻³
If two objects occupy the same volume, the one with lower density has the lower mass
Volume isn’t always given directly — it often needs to be calculated from a shape formula first
What Is Density?
Density describes how tightly mass is packed into a given volume. A block of foam and a block of steel the same size clearly don’t weigh the same — the steel has far more mass squeezed into that identical volume, so it has a much higher density.
Density equationρ = m⁄V
Where ρ (the Greek letter “rho”) is density, m is mass, and V is volume. If mass is measured in grams and volume in cm³, density comes out in g cm⁻³; if mass is in kilograms and volume in m³, density comes out in kg m⁻³.
Same volume, very different mass — the box with more particles packed in is the denser one
Working Out Volume
Density questions don’t always hand you the volume directly — sometimes you need to calculate it first from the shape of the object.
SphereV = 4⁄3 πr³
CubeV = d³
CylinderV = πr²l
🧭 Recipe: Converting Units for Density
Larger to smaller unit — multiply (e.g. 2 m = 2 × 100 = 200 cm)
Smaller to larger unit — divide (e.g. 500 g = 500 ÷ 1000 = 0.5 kg)
Squared or cubed conversions — square or cube the conversion factor too (1 mm³ = (1⁄1000)³ m³ = 1 × 10⁻⁹ m³; 1 cm³ = (1⁄100)³ m³ = 1 × 10⁻⁶ m³)
Check your final units match what the question is actually asking for
Quick recap: ρ = m/V. Higher density means more mass packed into the same volume. Always double-check your volume is in the right units before dividing.
WE 1
A metal bar has a mass of 12.4 kg and dimensions 25 mm × 90 mm × 400 mm. Calculate the density of the material, in kg m⁻³.
Step 1 — Calculate the volumeV = 25 × 90 × 400 = 900 000 mm³ = 9.0 × 10⁵ mm³Step 2 — Convert mm³ to m³1 mm³ = 1 × 10⁻⁹ m³ → V = 9.0 × 10⁵ × 10⁻⁹ = 9.0 × 10⁻⁴ m³Step 3 — Substitute into the density equationρ = m ÷ V = 12.4 ÷ (9.0 × 10⁻⁴)≈ 13 800 kg m⁻³ (3 s.f.)
WE 2
A spherical ball bearing has a radius of 8.0 mm and a mass of 0.017 kg. Calculate the density of the material it is made from, in kg m⁻³.
Step 1 — Convert radius to metresr = 8.0 mm = 8.0 × 10⁻³ mStep 2 — Calculate the volume of the sphereV = 4⁄3 π r³ = 4⁄3 × π × (8.0 × 10⁻³)³ ≈ 2.14 × 10⁻⁶ m³Step 3 — Substitute into the density equationρ = m ÷ V = 0.017 ÷ (2.14 × 10⁻⁶)≈ 7930 kg m⁻³ (3 s.f.)
💡 Top tips
Always check whether you need to calculate volume from a shape formula before you can use the density equation
Convert every length into the same unit before calculating a volume — mixing mm and cm is a common source of errors
When converting cubed units, remember to cube the conversion factor, not just multiply once
Density has no dependence on the shape or amount of a substance — it’s a property of the material itself
⚠ Common mistakes
Forgetting to convert volume into the correct units before dividing (e.g. leaving it in mm³ when kg m⁻³ is needed)
Applying a linear conversion factor to a cubed unit, instead of cubing the factor first
Mixing up mass and weight when reading values from a question
Using the wrong volume formula for the shape described — double-check sphere, cube and cylinder formulas before substituting
Up next: Temperature Scales — where we look at how the Kelvin and Celsius scales relate to one another.
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