Density from mass and volume:
formula and worked example

What density tells you about a material

Density is mass per unit volume. It lets you compare materials without worrying about the size of the sample: a pebble and a boulder of the same rock have the same density even though their masses are very different.

In practice, density is how you identify an unknown metal, check whether concrete was mixed correctly, decide whether an object will float, or convert between a volume you can measure and a weight you need to pay for or transport.

The formula and unit discipline

• Mass: how much matter the object contains, in kilograms. Use a scale reading, not a weight in newtons.
• Volume: the space it occupies in cubic metres, from geometry or from water displacement.

Because the result is in kg/m³, the inputs should be in kg and m³. Most errors happen at the volume step. A litre is 0.001 m³, so a 2-litre container is 0.002 m³, and a cubic centimetre is one millionth of a cubic metre.

Laboratory results are often given in g/cm³. Multiply by 1,000 to convert to kg/m³: water at 1 g/cm³ is 1,000 kg/m³. The conversion is exact because both units scale together.

A 5.4 kg block occupying 2 litres

1. 1Write the volume in cubic metres: 2 litres = 0.002 m³.
2. 2Divide mass by volume: 5.4 ÷ 0.002 = 2,700.
3. 3Attach the unit: 2,700 kg/m³.
4. 4Cross-check in g/cm³: 5,400 g ÷ 2,000 cm³ = 2.7 g/cm³, which matches.

The result sits close to the value commonly quoted for aluminium, so the sample could plausibly be that metal, though density alone cannot prove it.

Using the number to draw conclusions

Comparing the result with the density of water, about 1,000 kg/m³, gives a quick float-or-sink answer for a solid object: lower density floats, higher density sinks. A ship floats because its average density, including the air in its hull, is below that of water even though steel is not.

• Match against reference values to narrow down the material, remembering that alloys and temperature shift the number.
• Average density for hollow or composite objects is lower than the solid material's density.
• Differences of a few per cent can be measurement error, so repeat the readings.

Treat density as a clue rather than an identification. Several different materials sit close together, and porosity or voids lower the measured value.

How a change in mass moves the density

When mass rises by 10% to 5.94 kg and volume stays fixed, density rises by exactly 10% to 2,970 kg/m³. Direct proportionality holds because volume is in the denominator and unchanged.

A real sample behaves differently: adding mass usually adds volume as well, so density stays put. A density change therefore signals a change of material or composition, such as a trapped air pocket or a different alloy, not just more of the same stuff.

Mistakes that skew density results

• Entering volume in litres or cm³ into a formula that expects m³, giving an answer 1,000 or 1,000,000 times off.
• Using weight in newtons instead of mass in kilograms.
• Measuring volume of an object that traps air bubbles when submerged, so displacement reads too high.
• Comparing with a handbook density without noting temperature or purity.

Applying it to a real object

For a regular block, measure length, width and height in metres, multiply to get volume, and weigh it. For an irregular object, submerge it in a measuring jug and read the rise in water level in millilitres, which equals cm³.

Once you have density, the formula works in reverse: mass equals density times volume. A 0.5 m³ batch of a 2,700 kg/m³ material has a mass of 1,350 kg, which is the number that matters for lifting and transport.

Density also converts between the two ways bulk goods are sold. Sand, grain and liquids are often priced by volume but shipped by weight, so a density figure turns one into the other. The same material can be loose, packed or wet, and each state has its own density, so measure the state you will actually handle.