## What you need to know

Mass is a measure of how heavy something is and is usually measured in grams (g) or kilograms (kg). Volume is a measure of how much space something takes up and is usually measured in cubic centimetres ($\text{cm}^3$) or cubic metres ($\text{m}^3$). Density is a measure of how tightly the mass of an object is packed into the space it takes up, so we calculate this by dividing its mass by its volume.

$\text{density } = \dfrac{\text{mass}}{\text{volume}}$

As a result, the units we used to measure density are compound units (for more information, see here (https://mathsmadeeasy.co.uk/gcse-maths-revision/conversions-gcse-revision-and-worksheets/)). The unit of choice will depend on what units are used to measure the values you’re given. If, in a particular question, mass is measured in kilograms and volume is measured cubic metres, then the density we calculate would be measured “kilograms per cubic metre”, which is denoted either by $\text{kgm}^{-3}$ or $\text{kg/m}^3$. If the starting units were different, then the resulting unit of density would be different. Let’s see an example.

Example: An object has a mass of 570g and a volume of 2,280 $\text{cm}^3$. Calculate its density.

As stated, density is equal to the mass divided by the volume, so this will be our calculation, but what will the unit of our answer be? In the question, the mass is given in grams and the volume in cubic centimetres, so the density should be in grams per cubic metre, which is denoted either like $\text{gm}^{-3}$ or $\text{g/m}^3$.

$\text{Density } = \dfrac{570}{2,280} = 0.25\text{ g/m}^3$

This is the unit of density you’ll see most often but not always, so watch out for the units given in the question. In general, the units will also be in the form “[units of mass] per [units of volume]”.

On top of calculating density, you will also be expected to be able to rearrange this formula and use it to find mass (when given the density and volume) and volume (when given the density and mass). From now on, let $d$ be density, $V$ be volume, and $m$ be mass. Then, our original equation looks like:

$d = \dfrac{m}{V}$

If we multiply both sides of the equation by $V$, then swap the left-hand side and right-hand side, we get

$m = d \times V$

So, we see that we can calculate mass by multiplying the density by the volume. Furthermore, if we then divide both sides by $d$ and swap the left-hand side and right-hand side, we get

$V = \dfrac{m}{d}$

Therefore, we can calculate volume by dividing the mass by the density. It’s good practice rearranging this formula to get it in the form that you want, but a quick way to remember how to calculate one of these values using the other two is to refer to the triangle below.

The way to use this triangle is as follows. Take your finger and cover up the letter which represents the thing that you’re trying to calculate. Then, the triangle will tell you what to do with the other two quantities to get the value you want.

For example, if we want to calculate the density, then we construct this triangle and cover up the $d$ (since that’s what we want).

Then, we see that what’s left over is “$m$ over $V$”, or in other words, $m$ divided by $V$ will give us the density. Learning this triangle and how to use it makes your life a lot easier.

Example: A cat has volume $0.004 \text{ m}^3$ and density $980 \text{ kg/m}^3$. Calculate the mass of the cat.

We’re looking for mass, so, constructing the triangle and covering up the $m$, we get

Therefore, to calculate the mass we must multiply the density by the volume. So

$\text{mass } = 0.004 \times 980 = 3.92\text{ kg}$

Note: we know that the units are kilograms by looking at the units in the question.

## Example Questions

#### 1) A bottle is filled with 2 kg of olive oil which has a density of $0.925 \text{ g/cm}^3$. What is the volume of the olive oil to the nearest whole number?

We are looking for volume, so covering up the $V$ we see from the triangle above that we must divide $m$ by $d$.

Before we can do this, however, we must notice that the units in the question don’t line up. The mass is in kilograms, but the density is in grams per cubic centimetre. So, we must first convert the kilograms into grams before proceeding.

$2\text{ kg} = 2000 \text{ g}$

$\text{Therefore, volume } = \dfrac{2000}{0.925} = 2,162\text{ cm}^3$.

#### 2) A cube with side-length 7m has density $10,800,000 \text{ kg/m}^3$. Work out the mass of the cube.

We are looking for mass, so covering up the $m$ we see from the triangle above that we must multiply $d$ by $V$. As it happens, we don’t know the volume, but we do know that the shape is a cube with side-length 7m, so

$\text{Volume of cube } = 7 \times 7 \times 7 = 343\text{ m}^3$.

Now, we can multiply the density by the volume to get the mass.

$\text{Mass } = 343 \times 10,800,000 = 3,704,400,000 \text{ kg}$.

## Density Mass Volume Revision and Worksheets

Density and Pressure
Level 4-5
Speed and Density
Level 4-5

For all those Maths teachers out there looking for new density mass and volume worksheets and practice questions, you have arrived in the right place. It doesn’t matter if you are a Maths tutor in York or a GCSE Maths teacher in London, our density mass volume revision materials will be what you are looking for.