Density

What is Density?

Density is the amount of mass packed into a given volume. It tells you how tightly matter is squashed together in an object. A material with high density has a lot of mass in a small space; a low-density material has little mass in a large space.

Formula:

Density = Mass / Volume

D = M / V

The units depend on what units you use for mass and volume:

• kg/m³ (kilograms per cubic metre) — used in physics
• g/cm³ (grams per cubic centimetre) — common in chemistry and everyday problems

Water has a density of exactly 1 g/cm³ (or 1000 kg/m³). This is a useful reference point.

Rearranging the Formula

Like any three-variable formula, you can rearrange D = M / V to find any unknown:

• Mass = Density x Volume (M = D x V)
• Volume = Mass / Density (V = M / D)

A useful memory aid is the DMV triangle: write D on top, M and V on the bottom. Cover the unknown letter to see the correct formula — covering V gives M / D; covering M gives D x V.

Worked Example 1 — Finding Density

A metal cube has a mass of 540 g and a volume of 200 cm³. What is its density?

D = M / V = 540 / 200 = 2.7 g/cm³

This matches the density of aluminium.

Worked Example 2 — Finding Volume

A piece of gold has a mass of 193 g. Gold has a density of 19.3 g/cm³. What is its volume?

V = M / D = 193 / 19.3 = 10 cm³

Comparing Densities

Material	Density (g/cm³)
Air	0.0013
Wood (oak)	0.7
Water	1.0
Aluminium	2.7
Iron	7.9
Lead	11.3
Gold	19.3

Notice that metals are much denser than water, while wood is less dense. This single table explains why a coin sinks and a wooden plank floats.

Why Do Objects Float or Sink?

An object floats in a fluid (liquid or gas) if its density is less than the density of the fluid. It sinks if its density is greater than the fluid's density.

• Ice (0.92 g/cm³) floats in water (1.0 g/cm³) — this is why icebergs float, with about 90% submerged.
• Steel (7.9 g/cm³) sinks in water — yet a steel ship floats because the hull encloses a large volume of air, making the average density of the whole ship less than 1 g/cm³.
• A helium balloon floats in air because helium (0.00018 g/cm³) is much less dense than air (0.0013 g/cm³).

The key principle: compare the object's average density to the fluid's density.

Measuring Density in Practice

Regular shapes (cubes, cylinders, rectangular blocks): measure the dimensions with a ruler, calculate volume using a geometric formula (e.g. V = length x width x height), then weigh the object on a balance.

Irregular shapes — you cannot easily calculate the volume from measurements. Instead, use the displacement method (also called Archimedes' method):

1. Fill a measuring cylinder (or displacement can) with a known volume of water and record the level.
2. Carefully lower the object into the water using a thread.
3. Record the new water level.
4. The volume of water displaced equals the volume of the object.
5. Weigh the object and calculate D = M / V.

This method works because a submerged object displaces exactly its own volume of liquid — a discovery attributed to Archimedes, who reportedly leapt from his bath shouting "Eureka!" when he realised this.

Density and States of Matter

Solids are generally denser than liquids, which are denser than gases — because particles in a solid are packed tightly with little space between them, while gas particles are far apart with a great deal of empty space.

Water is a famous exception: liquid water (1.0 g/cm³) is denser than ice (0.92 g/cm³). This is why ice floats rather than sinks, and why lakes freeze from the top down — preserving aquatic life beneath the ice layer.

Key Terms

• Density — mass per unit volume (kg/m³ or g/cm³)
• Displacement — the volume of fluid pushed aside when an object is submerged
• Average density — total mass divided by total volume, accounting for hollow spaces
• Buoyancy — the upward force a fluid exerts on a submerged or floating object

Common Mistakes

• Confusing mass and weight — density uses mass (in kg or g), not weight.
• Using mismatched units — if mass is in grams and volume in m³, the calculation gives a very wrong answer. Convert units first.
• Assuming a heavier object is always denser — a large foam block may weigh more than a small iron nail, but the nail is far denser.
• Forgetting that ships float due to average density, not the density of steel alone.

Tips and Tricks

• Always write the unit alongside the answer: 2.7 g/cm³ is very different from 2700 kg/m³ (they are the same substance — just different units).
• The conversion: 1 g/cm³ = 1000 kg/m³.
• If an object's density is exactly equal to the fluid, it neither floats nor sinks — it remains suspended at whatever depth you place it.
• For the displacement method, make sure the object is fully submerged and does not touch the sides of the cylinder.