GCSE · Maths · AQA · Spec 8300 · Foundation

Compound units — density, pressure

A fake gold ring, an iron cannonball that floats, a high heel that out-presses an elephant. One idea explains all three: how much for just ONE unit?

Maths · Density

What does just ONE cm³ weigh?

Volume: 1/10. Volume 1 cm³. Mass 8 g. Density = mass ÷ volume 8 g/cm³. Mass ÷ density 1 cm³

Volume1 cm³Mass8 gDensity = mass ÷ volume8 g/cm³Mass ÷ density1 cm³

Here is a made-up metal where every 1 cm³ has a mass of 8 g. Drag the handle to change the volume of the block and watch the mass keep pace.

Exam line: Density is the mass of ONE unit of volume: density = mass ÷ volume. Mass in g and volume in cm³ give g/cm³; mass in kg and volume in m³ give kg/m³. To go the other way, mass = density × volume and volume = mass ÷ density.

Watch out: The density chip never moves. 1 cm³ weighs 8 g whether the block is tiny or big. Only the volume and the mass change, and they change together.

Maths · Comparing densities

Who floats, and who is a fake?

Drag each tag to its density in g/cm³: iron 7.86, mercury 13.6, the suspect ring 16 and gold 19.3. Then check, and read the line from left to right.

Place IronHg — MercuryPlace RingPlace Gold

Maths · Checking units

Spot the slip

Each block is a cuboid measuring 50 cm by 40 cm by 10 cm, made from a material with a density of 2000 kg/m³. A lorry can carry at most 500 kg. How many blocks can it carry?

A student's answer — which line goes wrong?

Density is in kg/m³, so I need the volume in m³ to get the mass in kg.
Volume of one block = 50 × 40 × 10 = 20 000 m³
Mass of one block = density × volume = 2000 × 20 000 = 40 000 000 kg
Number of blocks = 500 ÷ 40 000 000 = 0.0000125
Round down: 0 blocks. The lorry cannot carry even one.

Predict, then check

A force is a push or a pull, measured in newtons (N). Pressure is about how that push is spread out. Commit to a prediction first.

A person in high heels presses down through a heel of area 0.5 cm² with a force of 500 N. An elephant's foot has an area of 275 cm² and a force of 10 000 N. Which presses harder on the floor?

Maths · Pressure

Which area counts?

A solid block sits on a table. To find its pressure on the table you use pressure = force ÷ area. But a block has lots of areas: each face has one, and all the faces together make its total surface area.

Which area goes into the pressure calculation?

The total surface area: every face of the block added up.The biggest face, because that is the most area the block has.The area of the face in contact with the table: the base.

How sure are you?

I'm sureI think soI'm guessing

The total surface area: every face of the block added up.

This is a common misconception

Tempting, because 'surface area' is the area you have probably practised most. But the top and the sides are not pushing on the table at all. Only the part touching the table spreads the force, so dividing by the total surface area gives a pressure that is far too small.

The biggest face, because that is the most area the block has.

This is a common misconception

Size is not the test; touching is. If the block stands on a small face, the big face is up in the air and spreads nothing. Turn the block onto its big face and the contact area changes, and so does the pressure.

The area of the face in contact with the table: the base.

This matches the evidence

Yes. The force is spread over the area in contact, and it is the contact area that raises or lowers the pressure. The same force on a smaller contact area means a larger pressure.

Ask 'what is actually touching?' and divide the force by that area, the base in contact. Not the total surface area, and not just any face.

Worked example: backwards from pressure to a length

Problem

A cylindrical candle stands on a table. It presses down with a force of 15 N, and the pressure on the table is 0.5 N/cm². What is the radius of the candle's circular base?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A fake gold ring, a floating cannonball and a high heel that out-presses an elephant. One trick explains all three.

What you need to know

Density is the mass of ONE unit of volume, found by mass ÷ volume. Grams and cm³ give g/cm³; kilograms and m³ give kg/m³.
Rearrange it to find a mass (density × volume) or a volume (mass ÷ density), and compare densities to decide what floats or whether an object is genuine.
Pressure is the force on ONE unit of area, found by force ÷ area, in N/cm² or N/m². Use the area in contact, never the total surface area.
Check the units first and convert if you must, then give the answer in the right unit.

The big picture

Density and pressure are both 'how much for one unit': density is the mass in each unit of volume, and pressure is the force on each unit of area. Divide to get them, check the units match first, and use the area that is actually in contact.

Key points

1Density = mass ÷ volume; mass = density × volume; volume = mass ÷ density.

2Before calculating, match units: divide lengths in cm by 100 to get m; 1 tonne = 1000 kg; 1000 g = 1 kg.

3In a 'how many fit within a limit' problem: volume, then the mass of one item, then divide, and round DOWN.

4A density higher than another substance's means it sits further right on the scale: iron is below mercury, and a ring below gold is a fake.

5Pressure = force ÷ area; force = pressure × area; area = force ÷ pressure.

6A base area gives a length: square root for a square base, or solve πr² = area for a circular base.

.we-_R_18slubsnpfiv5ubrb_ [data-we-step]{display:flex !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-solution]{display:block !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-reveal]{display:none !important}

Worked example

Problem

A metal cube has a mass of 540 g and a volume of 200 cm³. (a) Find its density. (b) Find the mass of 350 cm³ of the same metal.

⚠ Watch out

Two slips cost the most. One is skipping the unit check: lengths in cm must become m before you find a volume in m³, or the density ends up in the wrong units. The other is dividing the force by the total surface area instead of the area in contact.

🧠

Memory hook

'Per' means divide. Density is mass PER cm³ and pressure is force PER cm². Heel versus elephant: squeeze the push onto a tiny area and the pressure shoots up.

✓

Check yourself

A 30 cm³ sample of a metal has a mass of 135 g. Find its density, with units. A box presses with 72 N on a base of 18 cm². Find the pressure, with units.

Flashcards

(14)

What is density, in plain words?

The mass of ONE unit of volume of a substance, for example how many grams are in each 1 cm³. (Mass is the quantity of matter in it.)

How do you calculate density, and where do its units come from?

density = mass ÷ volume. The units follow the ones you use: g and cm³ give g/cm³; kg and m³ give kg/m³.

Which two rearrangements of the density formula do you need?

mass = density × volume, and volume = mass ÷ density.

How does a ratio table find an object's mass from the density?

Row 1 pairs one unit of volume with its mass (the density). Row 2 pairs the object's volume with the missing mass. Scale row 1 by the same multiplier to reach row 2.

What must you check, and convert if needed, before calculating?

That the units of mass and volume match the units of the density. Lengths: cm ÷ 100 = m. 1 tonne = 1000 kg and 1000 g = 1 kg.

A cuboid problem asks how many items fit within a mass limit. What is the order of steps?

Volume first (length × width × height), then the mass of one item, then the limit ÷ the mass of one. Round DOWN so the limit is not exceeded.

Mercury has a density of 13.6 g/cm³ and iron 7.86 g/cm³. What follows?

Mercury's density is greater than iron's, so an iron cannonball floats in mercury.

How can a calculated density show whether an object is genuine?

Compare it with the known density. A ring of 16 g/cm³ is lower than gold's 19.3 g/cm³, so the ring is a fake.

What is pressure, and what is a force?

Pressure is the perpendicular force on a surface per unit of area. A force is a push or a pull, measured in newtons (N).

How do you calculate pressure, and what are its units?

Divide the force by the area. The units follow the ones you use: N/cm² for an area in cm², N/m² for an area in m².

Which area do you use for the pressure of a solid on a table or floor?

The area in contact, the base, not the total surface area.

Why can a high heel press harder than an elephant's foot?

Pressure is force per unit area. The heel's force is spread over a tiny area and the elephant's over a huge one: 1000 N/cm² against about 36.4 N/cm².

How do you find a force or an area from the pressure?

Multiply for the force (pressure × area); divide for the area (force ÷ pressure).

You have found a base area. How do you turn it into a length?

Square base (cube): the side is the square root of the area. Circular base (cylinder): solve πr² = area for r.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

Who floats, and who is a fake?

Spot the slip

Which area counts?

More AQA GCSE Maths topics