Density tells you how much mass is packed into a given volume. A bowling ball and a beach ball have almost the same size, but the bowling ball is far heavier because the material inside it is much denser.
Density also decides whether an object floats or sinks. This article gives the formula and units, shows how Archimedes' principle connects density with buoyancy, and lets you drop objects into different liquids to see the rule in action.
What is density?
Density is the mass of a material divided by the volume it occupies. It is a property of the material, not of the size of the sample: a pebble of iron and an iron beam have the same density.
Density
$latex \rho = \dfrac{m}{V}$
Here $latex \rho$ (the Greek letter rho) is the density, $latex m$ is the mass and $latex V$ is the volume. In the SI system the unit is $latex \dfrac{\text{kg}}{\text{m}^3}$. Chemistry and everyday work often use g/cm³, and the conversion is simple: $latex 1\ \dfrac{\text{g}}{\text{cm}^3} = 1000\ \dfrac{\text{kg}}{\text{m}^3}$. Liquid volumes in liters work too, since 1 L = 1000 cm³ = 0.001 m³.
The same formula can be rearranged to find the mass or the volume:
$$m = \rho \times V$$
$$V = \frac{m}{\rho}$$
When the object is a regular solid, compute its volume first. The articles on the volume of a cube, a cylinder and a sphere cover the formulas.
Densities of common materials
| Material | Density (kg/m³) | Floats in water? |
|---|---|---|
| Air | 1.2 | Yes (it is a gas) |
| Cork | 240 | Yes |
| Pine wood | 500 | Yes |
| Ice | 917 | Yes |
| Fresh water | 1000 | Reference |
| Sea water | 1025 | Reference |
| Aluminum | 2700 | No |
| Iron | 7870 | No |
| Lead | 11340 | No |
| Mercury | 13546 | No |
| Gold | 19300 | No |
These are typical values at room temperature. Wood varies a lot with species and moisture, and gases change with pressure and temperature.
Why do things float? Archimedes' principle
A fluid pushes up on anything placed in it. Archimedes' principle says the buoyant force equals the weight of the fluid displaced by the object:
Buoyant force
$latex F_b = \rho_{fluid} \times g \times V_{submerged}$
Here $latex g = 9.8\ \text{m/s}^2$ and $latex V_{submerged}$ is the part of the object that is below the surface. Two forces compete: the weight pulls down and the buoyant force pushes up.
- If the object is less dense than the fluid, it floats. It sinks only until the displaced fluid weighs as much as the object.
- If it is denser than the fluid, even full submersion gives a buoyant force smaller than the weight, so it sinks.
- If the densities are equal, it hovers wherever you leave it.
A floating object sinks to the point where the weights balance, which gives a clean result: the fraction of its volume under the surface equals the density ratio.
$$\frac{V_{submerged}}{V_{object}} = \frac{\rho_{object}}{\rho_{fluid}}$$
A steel ship floats because it is hollow. Its average density, counting the air inside the hull, is below the density of water.
Drop an object into a liquid
Choose a material or type your own density, pick a liquid and press the button. The arrows compare the weight of the object with the buoyant force. Try iron in mercury, then ice in sea water.
Drop an object into a liquid
Solved examples
Example 1
A metal block has a mass of 250 g and a volume of 100 cm³. Find its density in g/cm³ and in kg/m³.
$$\rho = \frac{m}{V} = \frac{250\ \text{g}}{100\ \text{cm}^3} = 2.5\ \dfrac{\text{g}}{\text{cm}^3}$$
Multiply by 1000 to convert: 2500 kg/m³. Compared with the table, the block is a bit less dense than aluminum, so it is probably an aluminum alloy.
Example 2
What is the mass of 2 liters of aluminum? Use $latex \rho = 2700\ \text{kg/m}^3$.
Convert the volume to cubic meters: $latex 2\ \text{L} = 0.002\ \text{m}^3$. Then use $latex m = \rho \times V$:
$$m = 2700 \times 0.002 = 5.4\ \text{kg}$$
The aluminum has a mass of 5.4 kg. The same volume of water has only 2 kg.
Example 3
A cube of ice ($latex 917\ \text{kg/m}^3$) floats in sea water ($latex 1025\ \text{kg/m}^3$). What fraction of the ice is under the surface?
$$\frac{V_{submerged}}{V_{ice}} = \frac{917}{1025} = 0.895$$
About 89.5% of the ice is submerged, which is why icebergs hide most of their volume. In fresh water the fraction would be 91.7%.
Example 4
A crown has a mass of 1 kg and displaces 62 cm³ of water when submerged. Is it pure gold ($latex 19.3\ \text{g/cm}^3$)?
The displaced water gives the crown's volume. Its density is:
$$\rho = \frac{1000\ \text{g}}{62\ \text{cm}^3} = 16.1\ \dfrac{\text{g}}{\text{cm}^3}$$
Pure gold would give $latex 1000 / 19.3 = 51.8$ cm³ of displaced water. The crown displaces more, so its density is lower than gold's and it is not pure gold. This is the story attributed to Archimedes.
Common mistakes
- Confusing density with weight. A bucket of water weighs more than an iron nail, yet the nail is far denser. Weight depends on how much material you have, density does not.
- Mixing units. Do not divide grams by cubic meters. Convert first: 1 g/cm³ = 1000 kg/m³.
- Using the whole volume in the buoyant force. For a floating object only the submerged part counts. The whole volume is used only when it is fully under the surface.
- Believing heavy objects sink. A loaded cargo ship is enormous and floats, while a small coin sinks. What matters is the average density.
- Ignoring hollow parts. Use the total volume, including empty spaces, when finding the density of the whole object.
Practice problems
Problem 1
A block has a mass of 0.5 kg and a volume of 0.0002 m³. What is its density?
Answer
$$\rho = \frac{0.5}{0.0002} = 2500\ \dfrac{\text{kg}}{\text{m}^3}$$
The density is 2500 kg/m³.
Problem 2
Ethanol has a density of 789 kg/m³. What is the mass of 0.5 L of ethanol?
Answer
$$m = 789 \times 0.0005 = 0.3945\ \text{kg}$$
0.5 L = 0.0005 m³, so the mass is 0.3945 kg, or about 394.5 g.
Problem 3
A material has a density of 8 g/cm³. Convert it to kg/m³ and say whether it floats in water.
Answer
8 g/cm³ = 8000 kg/m³.
Since 8000 is greater than 1000, the material sinks in water.
Problem 4
A cube of side 0.1 m and density 800 kg/m³ floats in fresh water. How deep is its base below the surface?
Answer
$$\frac{V_{submerged}}{V} = \frac{800}{1000} = 0.8$$
The base area is constant, so the submerged depth is $latex 0.8 \times 0.1 = 0.08$ m, or 8 cm.
Problem 5
Find the buoyant force on an object of volume 3 L that is fully submerged in sea water ($latex 1025\ \text{kg/m}^3$).
Answer
$$F_b = 1025 \times 9.8 \times 0.003 = 30.1\ \text{N}$$
The buoyant force is 30.1 N.
Problem 6
An object has a mass of 2 kg and a volume of 0.0025 m³. Does it float in water? If it does, what percentage is submerged?
Answer
$$\rho = \frac{2}{0.0025} = 800\ \dfrac{\text{kg}}{\text{m}^3}$$
Its density is less than 1000 kg/m³, so it floats with 80% of its volume under the surface.
Frequently asked questions
Is density the same as weight?
No. Weight is the force of gravity on an object, measured in newtons, and it grows with the size of the sample. Density does not depend on the size of the sample.
Why does ice float on water?
When water freezes its molecules arrange in an open structure that takes up about 9% more volume. The mass stays the same, so the density drops to 917 kg/m³, below liquid water.
Does density change with temperature?
Yes. Most materials expand when heated, so the same mass takes more volume and the density falls. Water is unusual: it is densest at about 4 °C.
How does a steel ship float if steel is denser than water?
The ship displaces a large volume of water because of its hollow shape. Once it displaces water whose weight equals the weight of the ship, the buoyant force balances the weight and the ship floats.
Related calculators
- Density calculator: find density, mass or volume from the other two.
See also
- Volume of a cube, volume of a cylinder and volume of a sphere: the volumes you need to find a density.
- Acceleration due to gravity: where the 9.8 in the weight and buoyancy formulas comes from.
- Examples of direct proportion: for one material, mass is directly proportional to volume.