Science

What is density, and what is the density formula?

Quick answer

Density is how much mass is packed into a volume. The formula is density = mass ÷ volume (ρ = m/V), measured in g/cm³ or kg/m³. Water's density is about 1 g/cm³.

Ages 11–16Secondary

In one sentence

Density tells you how tightly packed a material is: how much mass there is in each cubic centimetre (or cubic metre) of it.

Explained

Imagine two shoeboxes of exactly the same size. One is full of feathers, the other full of sand. The box of sand is far heavier, even though the boxes take up the same space. Sand has a much higher density than feathers: more mass is squeezed into the same volume.

density = mass ÷ volume, ρ = m / V

The symbol for density is ρ, the Greek letter rho (some books use D or d). Mass is measured in grams or kilograms (see mass and weight), and volume in cm³ or m³, so density has two common units:

g/cm³ (grams per cubic centimetre), handy for small objects in the lab. 1 cm³ is the same as 1 mL.
kg/m³ (kilograms per cubic metre), the SI unit used in physics.

To switch, use 1 g/cm³ = 1000 kg/m³. That is because 1 m³ holds 1,000,000 cm³, and 1,000,000 g is 1000 kg. Pure water has a density of about 1 g/cm³ = 1000 kg/m³, which makes it a handy reference.

The formula comes straight from the meaning: if 3 cm³ of a metal has a mass of 24 g, each cubic centimetre has 24 ÷ 3 = 8 g, so the density is 8 g/cm³. Rearranging gives the other two versions: m = ρ × V and V = m ÷ ρ. A formula triangle helps: write m on top and ρ and V underneath; cover the one you want.

boat
rock
balloon

A boat floats, a rock sinks, and a helium balloon rises: it all comes down to density.

Floating and sinking. An object floats in a liquid if it is less dense than the liquid and sinks if it is denser. Wood (about 0.5 g/cm³) floats on water; a stone (about 2.5 g/cm³) sinks. Oil floats on water, and a helium balloon rises because helium is far less dense than air. A steel ship floats even though steel is denser than water: the hull is full of air, so the ship as a whole (steel plus air inside) has a lower average density than water.

Why materials differ. In the particle model, density depends on how heavy the particles are and how closely they are packed. In solids and liquids the particles touch; in a gas they are far apart, so gases have very low densities (air is about 1.2 kg/m³). Heating usually makes a material expand, so its density goes down: that is why hot air rises. Water is the famous exception near freezing: ice is less dense than liquid water, so ponds freeze from the top down and fish survive below. More on this in solids, liquids and gases.

Density is a property of the material, not of the object: a paper clip and a steel girder have the same density. That is why measuring density can identify a material, or reveal a fake gold coin.

Examples

Material	Density (g/cm³)	Density (kg/m³)	In water it…
Air (room temperature)	0.0012	1.2	rises (it is a gas)
Pine wood (typical)	about 0.5	about 500	floats
Petrol	about 0.74	about 740	floats
Ice	0.92	920	floats
Cooking oil	about 0.92	about 920	floats
Water	1.00	1000	—
Seawater	about 1.03	about 1030	—
Aluminium	2.7	2700	sinks
Steel	about 7.8	about 7800	sinks
Silver	10.5	10,500	sinks
Mercury (a liquid metal)	13.6	13,600	sinks
Gold	19.3	19,300	sinks

Worked example: a block measures 5 cm × 4 cm × 2 cm and has a mass of 120 g. Volume = 5 × 4 × 2 = 40 cm³. Density = 120 ÷ 40 = 3 g/cm³, which is 3000 kg/m³.

Common mistakes

Dividing the wrong way round. It is mass ÷ volume, never volume ÷ mass. Fix: the units say it, grams per cm³.
Mixing units. Kilograms with cm³ gives a strange answer. Fix: use g with cm³, or kg with m³, and convert with 1 g/cm³ = 1000 kg/m³.
Thinking heavy means dense. A huge log is heavy but floats; a tiny gold ring is light but very dense. Density compares mass with volume.
Forgetting the container. When you weigh a liquid in a measuring cylinder, subtract the mass of the empty cylinder first.

Step by step

Measure the massUse a balance and write the mass in grams (or kilograms).
Find the volumeRegular shape: measure it and use a formula (a cuboid is length × width × height). Irregular object: lower it into a measuring cylinder of water; the rise in the water level is its volume. Liquid: read the volume off a measuring cylinder.
Check the unitsGrams go with cm³; kilograms go with m³.
DivideDensity = mass ÷ volume.
CompareIs it less than 1 g/cm³ (floats in water) or more (sinks)? Does it match a known material?

The water method for irregular objects is called displacement. A displacement can (eureka can) works the same way: the water that overflows into a cylinder has the same volume as the object.

Practice questions

Density calculations

Try each one, then check your answer.

An object has a mass of 60 g and a volume of 20 cm³. Find its density.
Show answer
60 ÷ 20 = 3 g/cm³
Find the density of 500 kg of material with a volume of 2 m³.
Show answer
500 ÷ 2 = 250 kg/m³
A metal block has a mass of 54 g and a volume of 20 cm³. Find its density.
Show answer
54 ÷ 20 = 2.7 g/cm³ (aluminium)
A cube has sides of 2 cm and a mass of 64 g. Find its density.
Show answer
V = 2 × 2 × 2 = 8 cm³; 64 ÷ 8 = 8 g/cm³
A block is 5 cm × 4 cm × 2 cm and has a mass of 120 g. Find its density.
Show answer
V = 40 cm³; 120 ÷ 40 = 3 g/cm³
Concrete has a density of 2400 kg/m³. What is the mass of 3 m³?
Show answer
m = ρ × V = 2400 × 3 = 7200 kg
Oil has a density of 0.92 g/cm³. What is the mass of 250 cm³ of oil?
Show answer
0.92 × 250 = 230 g
Gold has a density of 19.3 g/cm³. What is the volume of 386 g of gold?
Show answer
V = m ÷ ρ = 386 ÷ 19.3 = 20 cm³
Mercury has a density of 13.6 g/cm³. What volume does 1 kg of mercury take up?
Show answer
1000 ÷ 13.6 ≈ 73.5 cm³
Convert 2.7 g/cm³ into kg/m³.
Show answer
2.7 × 1000 = 2700 kg/m³
Convert 1.2 kg/m³ into g/cm³.
Show answer
1.2 ÷ 1000 = 0.0012 g/cm³
A 150 g stone is lowered into a measuring cylinder. The water rises from 50 cm³ to 110 cm³. Find the stone's density.
Show answer
V = 110 − 50 = 60 cm³; 150 ÷ 60 = 2.5 g/cm³
An empty measuring cylinder has a mass of 120 g. With 50 cm³ of a liquid in it, the mass is 160 g. Find the liquid's density.
Show answer
Liquid = 160 − 120 = 40 g; 40 ÷ 50 = 0.8 g/cm³
A block has a mass of 300 g and a volume of 600 cm³. Will it float in water?
Show answer
300 ÷ 600 = 0.5 g/cm³, less than water's 1 g/cm³, so yes, it floats.
Steel has a density of 7.8 g/cm³. What is its relative density (compared with water)?
Show answer
7.8 ÷ 1 = 7.8 (no unit)
A cylinder has radius 2 cm, height 5 cm and mass 170 g. Find its density.
Show answer
V = π × 2² × 5 = 20π ≈ 62.8 cm³; 170 ÷ 62.8 ≈ 2.7 g/cm³, probably aluminium

Word problems

Density in real life

Try each one, then check your answer.

A ring is sold as pure gold. It has a mass of 21 g and a volume of 2 cm³. Is it gold?
Show answer
21 ÷ 2 = 10.5 g/cm³. Gold is 19.3 g/cm³, so no: 10.5 g/cm³ matches silver.
A classroom is 8 m × 6 m × 3 m. Air has a density of 1.2 kg/m³. What is the mass of the air in the room?
Show answer
V = 144 m³; 1.2 × 144 = 172.8 kg, more than two adults!
A fish tank is 50 cm × 30 cm × 40 cm. What is the mass of the water when it is full?
Show answer
V = 60,000 cm³; at 1 g/cm³ that is 60,000 g = 60 kg
What is the mass of the oil in a 1-litre bottle? (oil: 0.92 g/cm³)
Show answer
1 L = 1000 cm³; 0.92 × 1000 = 920 g
Ice is 0.92 g/cm³ and seawater about 1.03 g/cm³. A floating iceberg has the fraction 0.92 ÷ 1.03 of its volume under water. About what percentage is that?
Show answer
0.92 ÷ 1.03 ≈ 0.89, so about 89%: roughly nine tenths of an iceberg is hidden.
A car's petrol tank holds 50 litres. Petrol is about 0.74 kg per litre. What is the mass of a full tank of petrol?
Show answer
0.74 × 50 = 37 kg

Questions people ask

What is the density formula?

Density = mass ÷ volume, ρ = m/V. Rearranged: mass = density × volume, and volume = mass ÷ density.

What are the units of density?

The SI unit is kg/m³. In school labs g/cm³ (the same as g/mL) is more common. 1 g/cm³ = 1000 kg/m³.

What is the density of water?

About 1 g/cm³, or 1000 kg/m³. It is very slightly less when water is warm and highest at about 4 °C. Seawater is a little denser, about 1.03 g/cm³, because of the dissolved salt.

How do you remember the density formula?

Use the triangle: m on top, ρ and V below. Cover what you want: covering ρ leaves m over V (divide), covering m leaves ρ next to V (multiply).

What is relative density?

The density of a material divided by the density of water. It has no unit: gold's relative density is 19.3, so gold is 19.3 times as dense as water. It is also called specific gravity.

Why does ice float?

When water freezes, its molecules lock into an open pattern that takes up about 9% more space, so ice (0.92 g/cm³) is less dense than liquid water and floats.

What is Density in Minecraft?

It is the name of an enchantment for the mace weapon that makes falling attacks hit harder. It borrows the word, but it is game vocabulary, not physics. The same idea of "how much per space" does appear elsewhere in school, for example population density (people per km²) in geography.

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