Which is heavier — a kilogram of feathers or a kilogram of lead? Neither; that’s the catch. The real difference is density: how tightly mass is packed into space. A bucket of feathers and a bucket of sand take up the same room, but one is far heavier. That “heaviness for its size” is exactly what density measures — and it’s just one short formula away.
📚 What you need to know
Density (ρ) is mass per unit volume: ρ = m/V.
SI unit is kg m−3; you’ll also meet g cm−3 (and 1 g cm−3 = 1000 kg m−3).
Same volume + more mass → higher density.
Volume often isn’t given — you compute it: cuboid l×w×h, cube d3, sphere 4/3πr3, cylinder πr2l.
The classic trap is converting volume units: cube the linear factor → 1 mm3 = 10−9 m3, 1 cm3 = 10−6 m3.
Handy benchmark: water ≈ 1000 kg m−3 (= 1 g cm−3); gases are about 1000× less dense.
What density really measures
Density answers one simple question: how much mass is squeezed into a given amount of space? Two objects can be the same size yet have very different masses — the heavier one is denser. Fill one box with sand and an identical box with feathers: same volume, but the sand box is far heavier, because sand has the higher density.
Densityρ = m / V
Same-sized boxes, different amounts of stuff: more mass in the same volume means a higher density.
Beware the classic trick question: “which weighs more, a kilogram of feathers or a kilogram of lead?” They’re equal — a kilogram is a kilogram. Density is the fair comparison, because it fixes the volume and asks about the mass, not the other way round.
The formula and its units
You’ll rearrange ρ = m/V constantly, so keep all three versions to hand:
Rearrangedρ = m/V · m = ρV · V = m/ρ
The units drop straight out of the formula. Put mass in kilograms and volume in cubic metres and the density comes out in kg m−3 — the SI unit. In chemistry you’ll often see g cm−3 instead, and the two are linked by 1 g cm−3 = 1000 kg m−3. Water is the number worth memorising: about 1000 kg m−3, or equivalently 1 g cm−3.
If a question hands you g cm−3, don’t panic. Either convert to kg m−3 (multiply by 1000), or keep everything in grams and centimetres and convert only the final answer. The one rule: pick a single unit system and stick with it for the whole calculation.
When the volume isn’t handed to you
Half the battle is that questions rarely give you the volume directly — they give you the shape and its dimensions and expect you to build it. These three shapes cover almost everything you’ll meet:
The three shapes you’ll use most, with the dimensions each volume formula needs.
Volume of common shapesCuboid V = l × w × h (cube: V = d3)Sphere V = 4/3 πr3Cylinder V = πr2l
One golden rule: put every length into metres before you square or cube it. Then your volume is already in m3, ready to divide straight into a mass in kg.
The unit trap that eats marks
This is where more marks are lost than anywhere else on the topic. When you convert a volume, you have to convert every dimension — which means you cube the conversion factor, not just the number in front.
A metre is 1000 mm along one edge, so a cubic metre is 10003 = 109 cubic millimetres. Flip it round: 1 mm3 = 10−9 m3.
So a millimetre is 10−3 of a metre, but a cubic millimetre is (10−3)3 = 10−9 of a cubic metre. In the same way, 1 cm3 = 10−6 m3. Miss the cube and your density comes out wrong by a factor of a thousand or a million.
If you ever blank on which power to use, the data booklet lists these conversions — but the safest habit is to convert every length to metres at the very start, so the volume is in m3 from the outset and this trap simply never appears.
Worked examples
WE 1
A rectangular block of aluminium measures 20 mm × 60 mm × 100 mm and has a mass of 324 g. Find its density in kg m−3.
Mass in kg: m = 324 g = 0.324 kgVolume of the cuboid: V = l × w × hV = 20 × 60 × 100 = 120 000 mm³ = 1.2 × 105 mm³Convert mm³ → m³ (× 10−9):V = 1.2 × 105 × 10−9 = 1.2 × 10−4 m³Density: ρ = m / Vρ = 0.324 ÷ (1.2 × 10−4) = 2700ρ = 2700 kg m−3Bang on aluminium’s real density — a reassuring sanity check.
WE 2
A steel ball bearing has a radius of 1.2 cm. Steel has a density of 7900 kg m−3. Find the mass of the ball bearing.
Rearrange ρ = m/V for mass: m = ρVRadius in metres: r = 1.2 cm = 0.012 mVolume of the sphere: V = 4/3 πr³V = 4/3 × π × (0.012)³ = 7.24 × 10−6 m³Mass: m = ρVm = 7900 × (7.24 × 10−6) = 0.0572 kgm ≈ 57 gA small ball, but steel is dense — about the mass of a large egg.
WE 3
A solid plastic cylinder has radius 3.0 cm and height 10 cm, with a mass of 0.25 kg. Find its density, and decide whether it floats in water (ρwater = 1000 kg m−3).
Lengths in metres: r = 0.030 m, l = 0.10 mVolume of the cylinder: V = πr²lV = π × (0.030)² × 0.10 = 2.83 × 10−4 m³Density: ρ = m / Vρ = 0.25 ÷ (2.83 × 10−4) = 884ρ ≈ 880 kg m−3 — it floats880 < 1000, so the plastic is less dense than water and floats — just like most plastics.
🔧 Nailing a density calculation
Mass in kg. If it’s in grams, divide by 1000.
Find the volume. If it isn’t given, pick the shape formula — and put every length in metres first.
Fix the volume units. If you worked in mm or cm, convert to m3 by cubing the factor (×10−9 or ×10−6).
Divide.ρ = m/V.
Sanity-check. Compare with water (1000) or a known metal — a wildly off answer usually means a units slip.
dimensions l, w, r, h …
shape formula
volume V in m³
m ÷ V
density ρ kg m−3
Quick recap: Density is mass per unit volume, ρ = m/V, in kg m−3 (or g cm−3). Same volume with more mass means higher density. When the volume isn’t given, build it from the shape — and watch the units: converting mm3 or cm3 to m3 means cubing the factor (10−9 and 10−6). Water sits at a handy 1000 kg m−3.
💡 Top tips
Go SI first. Mass in kg, lengths in m — then the density lands in kg m−3 automatically.
Cube the conversion, not just the number. 1 cm = 10−2 m, so 1 cm3 = (10−2)3 = 10−6 m3.
Keep a few benchmarks in your head: water 1000, aluminium 2700, steel ~7900, copper 8960, gold 19300, air ~1.2 (all kg m−3).
1 g cm−3 = 1000 kg m−3. A quick bridge between the two common unit systems.
Floats or sinks? Compare density with the liquid — less dense floats, denser sinks.
⚠ Common mistakes
Converting only the number, not cubing it — writing 1 cm3 = 10−2 m3 instead of 10−6 m3
Forgetting to change grams to kilograms before dividing
Using diameter instead of radius in the sphere or cylinder formula
Mixing units — mass in kg but lengths left in cm, so the density is out by a huge factor
Rounding the volume too early, then getting a density that’s noticeably off
Density ties a substance’s mass to the space it fills. Next we switch from “how much stuff” to “how hot it is” — and it turns out temperature is really a story about how fast those particles are jiggling. Physics measures it on its own scale, the kelvin. Coming up: temperature scales — kelvin vs Celsius, absolute zero, and the one conversion you’ll reach for again and again.
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