Density: Definition and Simple Calculations

Density is the mass of a substance per unit volume, ρ = m ÷ V, measured in kilograms per cubic meter (kg/m3) or grams per cubic centimeter (g/cm3). In brief: density is a property of the material, not of the size of the sample. 1 g/cm3 = 1,000 kg/m3. An object floats in a fluid if its average density is lower than that of the fluid. Rearranged: m = ρV and V = m ÷ ρ.

Lead feels heavy and polystyrene feels light even when the samples have the same size, because they have different densities. This guide defines density, shows how to use the formula, lists typical densities, explains floating and sinking, compares materials in a table, works through calculations and corrects common mistakes.

What is density?

Density tells how much matter is packed in a given volume: ρ = m ÷ V, where ρ (Greek letter rho) is density, m is mass and V is volume. Density depends on the mass of the individual atoms or molecules and on how closely they are packed. Gases have low densities because their particles are far apart, and solids and liquids have much higher densities. Density is an intensive property: a gold ring and a gold bar have the same density, but different masses.

Density depends on temperature and, for gases, on pressure. Water has a maximum density of 1,000 kg/m3 near 4 °C, and ice (917 kg/m3) is less dense than liquid water, which is unusual.

How is density calculated?

  1. Measure the mass with a balance, in kg or g.
  2. Find the volume: for a regular solid use a formula (length × width × height for a box), for a liquid use a measuring cylinder, and for an irregular solid use displacement.
  3. Divide: ρ = m ÷ V. Use consistent units: kg and m3 or g and cm3.

The density triangle helps with the rearrangement: m = ρ × V and V = m ÷ ρ. Relative density (specific gravity) is the ratio of the density of a substance to that of water, and has no unit.

Mass against volume for water, aluminum and iron 0 2 4 6 8 10 0 20 40 60 80 Volume (cm³) Mass (g) iron 7.87 g/cm³ aluminum 2.7 water 1.00
Mass against volume for three materials. Each graph is a straight line through the origin, because mass = density × volume. The gradient is the density: the steeper the line, the denser the material (iron 7.87, aluminum 2.7 and water 1.00 g/cm³).

How does density determine floating?

An object placed in a fluid floats if its average density is lower than the density of the fluid, and sinks if it is higher. A steel ship floats because its hull encloses a lot of air, so the average density of the whole ship is lower than that of water. Oil floats on water (oil density about 900 kg/m3), and a hot air balloon rises because the hot air inside is less dense than the surrounding air.

Densities of common materials

Approximate densities (at room temperature and normal pressure)
Material kg/m3 g/cm3
Air1.20.0012
Wood (pine)about 500about 0.5
Ice9170.917
Water1,0001.00
Aluminum2,7002.7
Iron7,8707.87
Lead11,34011.3
Gold19,30019.3

Worked example: density calculations

  1. Find the density. A metal block measures 20 cm × 10 cm × 5.0 cm and has a mass of 2.70 kg. V = 20 × 10 × 5.0 = 1,000 cm3 = 1.00 × 10−3 m3. ρ = 2.70 ÷ 1.00 × 10−3 = 2,700 kg/m3, which matches aluminum.
  2. Find the mass. The mass of 2.5 L of petrol with a density of 0.74 g/cm3: V = 2.5 L = 2,500 cm3; m = ρV = 0.74 × 2,500 = 1,850 g = 1.85 kg.
  3. Find the volume. The volume of 500 g of gold: V = m ÷ ρ = 500 ÷ 19.3 = 25.9 cm3, about the size of a small cube with 3 cm sides.
  4. Will it float? A wooden block has a mass of 120 g and a volume of 200 cm3. ρ = 120 ÷ 200 = 0.60 g/cm3, which is less than 1.00 g/cm3 for water, so it floats. In oil (0.90 g/cm3), it still floats.
20 cm 10 cm 5.0 cm m = 2.70 kg V = 1.00 × 10−3 m3 ρ = 2,700 kg/m3

How was density first used?

According to legend, Archimedes used density in the third century BCE to find out whether a crown was pure gold, by measuring the volume of the crown with the water it displaced. Modern uses include checking the purity of precious metals, the concentration of battery acid and sugar solutions with a hydrometer, and calculating the buoyancy of ships and balloons.

Which mistakes are common?

Mistake 1: Mixing units

Do not combine grams with m3 or kilograms with cm3 without conversion.

Mistake 2: Thinking that bigger objects are denser

Density is independent of the size of the sample.

Mistake 3: Confusing density with weight

Density is mass per volume. Weight is the gravitational force.

Mistake 4: Saying that all heavy objects sink

What matters is the average density compared with the fluid. A large ship floats, and a small steel ball sinks.