Density: ρ = m/V, unit kg/m³
Density is mass per unit volume: ( in kg, in m³, in kg/m³; g/cm³ also accepted). Two elements score the definition mark: the ratio "mass per unit volume" AND the unit. "Per" means divided by — "mass × volume" scores zero. Water ≈ 1000 kg/m³ (1.0 g/cm³): an object less dense than water floats, denser sinks. Rearrange: and . Mini-example: 350 g in 18 cm³ gives g/cm³.
Three ways to measure density
Regular solid: mass on a balance, volume from a ruler (, or ), then . Irregular solid (displacement): mass on a balance; lower it on a thread into a filled displacement (eureka) can; collect the overflow in a measuring cylinder; that volume = the solid's volume. Liquid: mass of empty cylinder ; add volume ; mass of cylinder + liquid ; . Read every volume at eye level.
1 g/cm³ = 1000 kg/m³
The two density units are kg/m³ (SI) and g/cm³, with (since 1 g kg and 1 cm³ m³). Common values: water 1.0 g/cm³ = 1000 kg/m³; aluminium ≈ 2.7 g/cm³ = 2700 kg/m³; iron ≈ 7.9 g/cm³ = 7900 kg/m³. Cambridge advises keeping g and cm³ when the data is given that way, then multiplying by 1000 only if kg/m³ is required. Always state the unit the question asks for.
Drawn from real examiner reports.
"Per" means divided by, not times
"Mass per unit volume" means mass ÷ volume — "per" signals a ratio. Candidates who read "per" as "multiplied by" pick , which gives kg·m³, not kg/m³. Think: "per unit volume" is the mass shared across every 1 m³ of material. In an MCQ, do not choose the largest-looking option — check the unit reads kg/m³ or g/cm³. Correct form: .
November 2023 Paper 11 MCQ: "Many candidates selected option A indicating that many thought that 'per' means 'multiplied by' rather than 'divided by', in spite of the equation immediately following the definition."
Inverting the formula: V/m instead of m/V
Dividing volume by mass () gives m³/kg — the reciprocal of density (specific volume), not density. For a steel sample of mass 350 g and volume 18 cm³: correct g/cm³; wrong — far too small and the wrong unit. Before choosing an MCQ option, sense-check the magnitude: a metal is 5–20 g/cm³, never a fraction below 1.
June 2024 Paper 11 MCQ Q3: "weaker candidates had not rearranged the density equation correctly and instead divided volume by mass" — options C and D were the most common incorrect responses. June 2024 Paper 42/43: "the most common errors involved an incorrect rearrangement of the density equation."
A density answer needs a unit
A missing or wrong unit loses the accuracy mark. The usual slips: writing "g" for "g/cm³", "kg" for "kg/m³", or "g/m³" (a mixed unit, wrong by a factor of ), or leaving a bare number. Density is always mass ÷ volume, so its unit is kg/m³ or g/cm³ — never omit it.
November 2023 Papers 51–53 practical: "some omitted a unit". June 2024 Papers 42/43: "a number of candidates gave an incorrect unit for density." November 2024 Paper 31 mark scheme: a missing or incorrect unit normally means the final calculation mark is not awarded.
Mixing g, cm³ and SI units
Work entirely in g and cm³ (answer g/cm³) or entirely in kg and m³ (answer kg/m³) — never mix. Substituting mass in kg but volume in cm³, or grams ÷ m³, throws the answer out by or . Cambridge advises keeping the data in g and cm³ and finding the mass in grams to avoid powers-of-ten errors.
November 2023 Paper 31: candidates "substituted the mass in kilograms rather than grams". November 2023 Paper 42: it is simpler to leave the volume in cm³ and density in g/cm³ and calculate the mass in grams.
Density uses mass, not weight
Density is — mass ÷ volume. Substituting weight (N) instead of mass (kg) gives N/m³, which is not a density. Use a balance reading (mass, kg), not a newton-meter reading (weight, N). Choose the right volume tool too: a displacement can for an irregular solid, a measuring cylinder for a liquid.
Practical density: 3 mark stages
P5/P6 density questions award separate marks for: naming the instrument for mass (a balance); describing how the volume is found (displacement can + measuring cylinder for an irregular solid); and showing the density calculation. Name each stage explicitly.
Name a measurement error, not "poor practice"
For "state a source of error", name a specific measurement inaccuracy — reading the meniscus from above, or water not level with the overflow spout — and say how it changes the density. General answers like "not reading carefully" or "human error" earn nothing.
Four-step density calculation
For any density calculation: (1) write or the rearranged form; (2) check units — kg and m³, or stay in g and cm³ consistently; (3) substitute and evaluate; (4) state the unit. Writing the equation first earns the method mark even if the arithmetic slips.
| Quantity | Symbol | Formula | SI Unit |
|---|---|---|---|
| Density | kg/m³ | ||
| Mass | kg | ||
| Volume | m³ |
Where: = density (kg/m³), = mass (kg), = volume (m³).
Unit conversion:
Key reference value: density of water (or )
Define density.
A piece of metal has a mass of 350 g and a volume of 18 cm³.
Calculate the density of the metal. Give your answer in g/cm³.
