Press the flat pad of your thumb hard against the point of a drawing pin. Now press the pin the other way round, with the sharp point against your thumb, using exactly the same push. The first way barely marks you; the second way hurts at once and could pierce the skin. Same force, same thumb, wildly different result — so what changed?

What changed is the area the force was squashed into. The blunt head spreads your push across a wide patch of skin; the needle-sharp point crams the identical push into a speck of an area. That "how concentrated is the force" idea is called pressure, and it decides whether a knife slices, whether snowshoes stop you sinking, and — once we get to fluids — whether a boat floats.

The pressure formula

Pressure measures how much force is pushing on each little bit of area. To turn that into a number we simply divide the force by the area it is spread over:

Here F is the force in newtons (N) and A is the area it presses on in square metres (\text{m}^2). Because it is newtons divided by square metres, pressure is measured in pascals: one pascal is one newton spread over one square metre.

The formula tells you the whole story in one glance. For a fixed force, shrinking the area makes the pressure shoot up (a sharp point), while spreading the force over a big area makes the pressure tiny (lying flat on ice instead of standing on it).

Worked example 1 — finding the pressure

A brick lies on a table. It presses down with a force (its weight) of 18\ \text{N}, and the face resting on the table has an area of 0.03\ \text{m}^2. What pressure does it put on the table?

Step 1 — write the formula.

Step 2 — put the numbers in.

Step 3 — check it makes sense. Stand the very same brick up on its small end (say 0.01\ \text{m}^2) and the same 18\ \text{N} now gives 18/0.01 = 1800\ \text{Pa} — three times the pressure, from the very same brick, just because the area is three times smaller.

Worked example 2 — finding the force, then the area

Finding a force. A hydraulic press applies a pressure of 200\,000\ \text{Pa} onto a metal plate of area 0.05\ \text{m}^2. What force does it deliver? Here we know p and A and want F, so use F = p\,A:

Finding an area. A person of weight 600\ \text{N} wants to keep the pressure on soft snow down to just 3000\ \text{Pa} so they do not sink in. What total area of snowshoe do they need? Now rearrange to A = F/p:

That is why snowshoes are so wide: to reach the large area that keeps the pressure low. Their own bare feet, at maybe 0.04\ \text{m}^2, would give 600/0.04 = 15{,}000\ \text{Pa} — five times more — and down they'd plunge.

Sharp means small area means high pressure

Once you have p = F/A in your head, a whole shelf of everyday things suddenly makes sense — and they split neatly into two camps.

• Make a small area on purpose → high pressure. A knife is sharpened to a razor-thin edge so a gentle push becomes a huge pressure that slices through. A drawing pin, a nail, a needle, an ice skate's blade and an animal's fang all work the same way: tiny area, enormous pressure.
• Make a big area on purpose → low pressure. Snowshoes, skis, a camel's broad feet, a tractor's fat tyres and the wide caterpillar tracks of a tank all spread the weight out so the pressure on soft ground stays low enough not to sink in.

Fluids push too: upthrust

Pressure isn't only about solids pressing on solids. A fluid — a liquid or a gas — presses on everything touching it, and here is the key fact: the deeper you go, the greater the pressure, because there is more fluid stacked up above weighing down. Dive to the bottom of a swimming pool and you can feel the extra pressure squeezing your ears.

Now think about an object sitting in water. The water pushes on it from every side. But the pressure on the bottom face is bigger than the pressure on the top face, simply because the bottom is deeper. A bigger push up on the bottom than down on the top leaves a leftover upward force. That net upward push from the fluid is upthrust (grown-ups also call it buoyancy). It is the reason things feel lighter in water, and the reason anything floats at all.

This builds directly on floating and sinking: there we said "the water pushes up"; now we know why — deeper water pushes harder, so a submerged object always gets a net shove upwards.

How big is the upthrust? Archimedes' principle

You might expect the up-push to be some complicated thing. It is astonishingly simple. When an object is put into a fluid it shoves some fluid out of the way — it displaces it. Archimedes discovered that:

Picture lowering a block slowly into water. As it sinks in, it pushes more and more water aside, so the upthrust grows. One of two things then happens:

• If the block can push aside its own weight of water before it is fully under, it stops sinking and floats, sitting with just enough of itself submerged to balance its weight.
• If even the fully-submerged block displaces less than its own weight of water, the upthrust can never catch up with the weight, so the block keeps going and sinks to the bottom.

Why a steel ship floats but a steel bolt sinks

This is exactly the same puzzle you met with density. A small steel bolt dropped in the sea sinks instantly. Yet a colossal steel ship, thousands of times heavier, floats. Archimedes explains it perfectly.

The bolt is a tiny solid lump. Even fully underwater it shoves aside only a bolt-sized splash of water — far, far less than its own weight — so the upthrust is feeble and the weight wins. The ship is spread out wide and hollow, with a vast belly of air. It only has to sink in a little before it has displaced an enormous amount of water — enough to weigh as much as the whole ship — and at that point the upthrust balances the weight and it floats. Same steel; utterly different displaced volume.

For a plain solid lump this is the same as the density rule: an object floats if it is less dense than the fluid and sinks if it is denser. A hollow boat wins because trapping air inside makes the whole boat-plus-air roomier — and so less dense on average — than the water around it.

See the tug-of-war

Below is a block in a tank of liquid. The down arrow is its weight; the up arrow is the upthrust from the liquid. Slide Block density below 1 (lighter than water) and the block bobs up and floats — it sinks in only far enough to displace its own weight, so the two arrows match. Push the density above 1 (denser than water) and even the fully-submerged block can't displace enough: the weight arrow beats the upthrust and it drops to the floor. The Block size slider grows both the block and, while it floats, the upthrust with it — a floating block always settles so upthrust equals weight.