What is the formula for density in GCSE Maths?

Use $\text{density} = \frac{\text{mass}}{\text{volume}}$. Match the units before substituting: grams divided by cubic centimetres gives density in $\text{g/cm}^3$.

How do I know whether to multiply or divide in a density question?

Identify the missing quantity first. For mass, multiply density by volume. For volume, divide mass by density. For density, divide mass by volume.

How do I convert density from grams per cubic centimetre to kilograms per cubic metre?

Multiply the numerical value by $1000$. There are $1{,}000{,}000\text{ cm}^3$ in $1\text{ m}^3$ and $1000\text{ g}$ in $1\text{ kg}$, so $1\text{ g/cm}^3 = 1000\text{ kg/m}^3$.

Does a heavier object always have a greater density?

No. Density depends on both mass and volume. A heavier object can have a lower density if its volume is sufficiently larger. Calculate mass divided by volume rather than comparing masses alone.

Common Mistakes with Density in GCSE Maths

Common mistakes with density in GCSE Maths

You recognise the question, remember the formula and reach for your calculator. Yet the answer still comes back wrong. The common mistakes with density in GCSE Maths usually involve dividing the wrong way round, mixing units or using the wrong volume. Fixing those decisions matters more than calculating faster.

Imagine a student reviewing a density question they got wrong. Their calculator work is accurate. The problem happened before they pressed a button: they used kilograms where the answer required grams.

That is useful information. Rather than revising everything again, they can identify the exact error and practise correcting it. This guide follows that approach, with worked examples and questions you can try yourself.

Your density checklist

Before calculating, ask:

Am I finding density, mass or volume?
Have I written the appropriate formula?
Do the units match the required answer?
Have I calculated volume rather than area?
Have I kept full accuracy until the final answer?

Density describes mass per unit volume. The three useful forms are:

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

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

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

If you need to revisit the explanation, the MathsGenie Speed and Density lesson brings the topic together.

Common mistakes and how to correct them

Dividing the wrong way round

Our imagined student sees two numbers and divides the larger by the smaller. This sometimes produces the correct answer, which makes the habit difficult to notice. But number size does not determine the operation. The quantity you need does.

The question: A material has mass $g540\text{ g}$ and density $g/cm32.7\text{ g/cm}^3$ . Find its volume.

The mistake: Multiplying the two values because the student remembers that density questions involve multiplication.

The correction: Write the formula for the missing quantity first:

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

Substitute, then calculate:

\frac{540}{2.7} = 200\text{ cm}^3

Finally, check backwards. Multiplying $cm3200\text{ cm}^3$ by $g/cm32.7\text{ g/cm}^3$ gives the original mass, $g540\text{ g}$ .

The repeatable method is simple: identify the unknown, choose the formula, match the units, substitute and check.

Practise that same sequence through MathsGenie's Speed and Density Lessons and Question Bank, so you can focus on correcting your method rather than finding suitable questions.

Mixing kilograms and grams

The next error is quieter. The formula is correct, but the units do not belong together.

The question: An object has mass $kg1.8\text{ kg}$ and volume $cm3600\text{ cm}^3$ . Find its density in $g/cm3\text{g/cm}^3$ .

The mistake: Calculating $1.8 \div 600$ and labelling the result $g/cm3\text{g/cm}^3$ .

That calculation uses kilograms, not grams. Writing a different unit afterwards does not convert the answer.

The correction: Read the requested unit before substituting. Convert the mass:

g1.8\text{ kg} = 1800\text{ g}

Then calculate:

g/cm3\text{density} = \frac{1800}{600} = 3\text{ g/cm}^3

The word “per” helps here. Grams per cubic centimetre means grams divided by cubic centimetres. Your numerator and denominator should match that description.

A hand-drawn stick-figure comic about a student weighing their calculator to check the weight of their density mistakes.

Using area instead of volume

Sometimes the question gives dimensions rather than volume. Our student multiplies two lengths, gets a convincing number and moves straight to density.

The missing third dimension changes everything.

The question: A solid cuboid measures $cm8\text{ cm}$ by $cm5\text{ cm}$ by $cm3\text{ cm}$ and has mass $g324\text{ g}$ . Find its density.

The mistake: Using $\times 5 = 40$ as the volume. That is the area of one face, measured in $cm2\text{cm}^2$ .

The correction: Calculate the cuboid's volume using all three dimensions:

\times 5 \times 3 = 120\text{ cm}^3

Now use the density formula:

g/cm3\text{density} = \frac{324}{120} = 2.7\text{ g/cm}^3

For other prisms, find the cross-sectional area and multiply by the prism's length. If that first stage is causing trouble, revisit volume of a prism before tackling more density questions.

Treating cubic conversions like length conversions

This density mistake often survives even when ordinary conversions feel straightforward.

Because $cm1\text{ m} = 100\text{ cm}$ , a student might assume the same conversion factor applies to cubic units. But a cube has three dimensions, and each dimension changes by that factor.

cm31\text{ m}^3 = 100^3\text{ cm}^3 = 1{,}000{,}000\text{ cm}^3

The question: Convert $g/cm32.7\text{ g/cm}^3$ into $kg/m3\text{kg/m}^3$ .

The mistake: Converting grams to kilograms but ignoring the volume unit underneath.

The correction: Think about a whole cubic metre. It contains $cm31{,}000{,}000\text{ cm}^3$ , so its mass is:

2.7 \times 1{,}000{,}000 = 2{,}700{,}000\text{ g}

Convert that mass into kilograms:

kg2{,}700{,}000\text{ g} = 2700\text{ kg}

Therefore:

kg/m32.7\text{ g/cm}^3 = 2700\text{ kg/m}^3

Explaining what happens to a whole cubic metre is safer than guessing whether to multiply or divide.

Rounding before the calculation is finished

A final answer may need rounding. An intermediate value usually does not.

The question: A sample has mass $g50\text{ g}$ and volume $cm37\text{ cm}^3$ . Find the mass of $cm312\text{ cm}^3$ of the same material, to three significant figures.

The mistake: Rounding the density to $g/cm37.1\text{ g/cm}^3$ first. That gives a mass of $g85.2\text{ g}$ .

The correction: Keep the density as an exact fraction during the calculation:

g\text{mass} = \frac{50}{7} \times 12 = \frac{600}{7}\text{ g}

Only round the final result:

figures\text{mass} = 85.7\text{ g}\quad\text{to three significant figures}

Keep the full calculator value, or use a fraction, until you reach the answer the question requests.

Exam-style density practice

Try these without looking at the answers. Write the formula and any conversion before calculating.

Find density: A block has mass $g960\text{ g}$ and volume $cm3120\text{ cm}^3$ . Find its density in $g/cm3\text{g/cm}^3$ .
Find mass: A liquid has density $g/cm30.8\text{ g/cm}^3$ . Find the mass of $cm3350\text{ cm}^3$ in grams.
Find volume: An object has mass $kg2.4\text{ kg}$ and density $g/cm33\text{ g/cm}^3$ . Find its volume in $cm3\text{cm}^3$ .

Check your answers

For the block:

g/cm3\text{density} = \frac{960}{120} = 8\text{ g/cm}^3

For the liquid:

g\text{mass} = 0.8 \times 350 = 280\text{ g}

For the object, first convert $kg2.4\text{ kg}$ into $g2400\text{ g}$ :

cm3\text{volume} = \frac{2400}{3} = 800\text{ cm}^3

For another targeted question, try this density and volume question. The Speed and Density exam booklet offers further practice.

When checking, label the cause of each error: formula, units, volume or rounding. Then redo the question without copying. A corrected answer shows what went wrong; a successful second attempt checks whether your method has changed.

Turn corrections into a reliable method

Our imagined student no longer treats every wrong density answer as the same problem. You can make that distinction too: knowing the formula is the starting point, while choosing it correctly and checking units takes deliberate practice. Start correcting your density mistakes with MathsGenie's Lessons and Question Bank, using the checklist on every attempt.

Common mistakes with density in GCSE Maths

Your density checklist

Before calculating, ask:

Am I finding density, mass or volume?
Have I written the appropriate formula?
Do the units match the required answer?
Have I calculated volume rather than area?
Have I kept full accuracy until the final answer?

Density describes mass per unit volume. The three useful forms are:

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

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

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

If you need to revisit the explanation, the MathsGenie Speed and Density lesson brings the topic together.