Density, Mass and Volume

Density tells you how heavy a material is for its size. It is the mass of each cubic centimetre of the material. Water has a density of about 1\text{ g/cm}^3, so each cubic centimetre of water has a mass of about 1 gram. Iron has a density of about 7.9\text{ g/cm}^3, so a cubic centimetre of iron is nearly 8 times as heavy.

A small bead and a large block of the same material have the same density. The block has a greater mass and a greater volume, but each cubic centimetre is just as heavy. So a piece cut from a block has the same density as the whole block.

In the exam you are given two of density, mass and volume, and asked to work out the third, for 2 or 3 marks near the end of the paper. Questions worth 3 or 4 marks make you work out a volume first, change kilograms into grams, or mix two materials together. Many students lose marks by multiplying when they should divide, so learn the formula well.

A metal key has a mass of 32 g and a volume of 4\text{ cm}^3. Share the 32 g between the 4 cubic centimetres: each one has 32\div4=8 g. So the density of the metal is 8\text{ g/cm}^3

To find a density, divide the mass by the volume:

\text{density}=\frac{\text{mass}}{\text{volume}}

The units come from the division. A mass in grams divided by a volume in \text{cm}^3 gives a density in grams per cubic centimetre, written \text{g/cm}^3. A mass in kilograms divided by a volume in \text{m}^3 gives kilograms per cubic metre, written \text{kg/m}^3

To compare two materials, work out both densities. The heavier object is not always made of the denser material. A large log is much heavier than a small coin, but the metal in the coin is far denser than the wood.

If each cubic centimetre of a metal has a mass of 8 g, then 5\text{ cm}^3 of it has a mass of 8\times5=40 g. So multiply to find a mass, and divide to find a volume:

\text{mass}=\text{density}\times\text{volume}

\text{volume}=\frac{\text{mass}}{\text{density}}

Cover the one you want to find. Covering M leaves D and V side by side, so multiply. Covering D leaves M over V, so divide the mass by the volume. Covering V leaves M over D, so divide the mass by the density.

Then check that your answer makes sense. When the density is more than 1\text{ g/cm}^3, the mass in grams is a bigger number than the volume in \text{cm}^3. When the density is less than 1\text{ g/cm}^3, the mass is the smaller number.

A density in \text{g/cm}^3 needs the mass in grams and the volume in \text{cm}^3. If a mass is given in kilograms, change it into grams first. There are 1000 grams in a kilogram, not 100, so multiply by 1000: 2.5 kg is 2500 g.

If the question asks for a mass in kilograms, work it out in grams, then divide by 1000 at the end. For example, 4700 g is 4.7 kg.

Large volumes are sometimes given in cubic metres. One cubic metre is 100\times100\times100=1\,000\,000\text{ cm}^3, and the question usually tells you this. Change the volume into \text{cm}^3 before you use a density in \text{g/cm}^3.

Read the last line of the question again before you round. For the greatest number of objects that can be carried, round down, as part of an object would go over the limit. For the number of bags you need to buy, round up, as one bag fewer would not be enough.

Many questions show you a solid and its measurements instead of giving its volume. Work out the volume first, then use it in the density formula.

• For a cuboid, multiply the length, the width and the height.
• For a prism, multiply the area of the cross-section by the length.
• For a cylinder, work out \pi r^2h, where r is the radius and h is the height.

Keep every figure of the volume on your calculator. Round only the final answer, because rounding the volume part of the way through can make the final answer wrong.

You can also work backwards. If you know the mass and the density of a cuboid, divide to find its volume. Then divide the volume by length \times width to find the height. If a container is filled with one thing and then with another, the volume is the same both times.

When two materials are mixed, the mixture has its own density. Find the total mass and the total volume, then divide:

\text{density of mixture}=\frac{\text{total mass}}{\text{total volume}}

Use the density formula on each part to find the mass or the volume you are not given. Each part keeps its own density, so use the right density for each one.

For example, 10\text{ cm}^3 of a metal with density 8\text{ g/cm}^3 has a mass of 80 g, and 30\text{ cm}^3 of a metal with density 4\text{ g/cm}^3 has a mass of 120 g. Together, 200 g fills 40\text{ cm}^3, so the density is 200\div40=5\text{ g/cm}^3

Don't add the two densities, and don't find their mean. The mean of 8 and 4 is 6, not 5, because there is more of the less dense metal.

• Multiplying the mass by the volume to find a density. Density is mass divided by volume.
• Dividing when you should multiply. To find a mass, multiply the density by the volume.
• Cubing the numbers because the unit is cubic centimetres. The small 3 belongs to the unit, so leave the numbers as they are.
• Using 100 to change between grams and kilograms. There are 1000 grams in a kilogram.
• Finding the mean of two densities. For a mixture, divide the total mass by the total volume.