Dive to the bottom of the deep end of a swimming pool and something odd happens: your ears begin to ache, pushed on from every side, and the deeper you go the harder that squeeze gets. Come back up and the ache fades. You have just felt pressure in a liquid — one of the most useful ideas in all of physics, and the reason dam walls are wedge-shaped, why a deep-sea submarine needs a hull like a tank, and why a burst pipe on the ground floor gushes harder than one in the attic.

In our lesson on pressure and upthrust we met pressure as force squeezed into an area, p = F/A, and saw that a fluid pushes harder the deeper you go. This page pins that down to a single, powerful formula and shows you exactly why the deep end pushes so much harder than the shallow end.

The squeeze comes from the weight of liquid above

Imagine a single point somewhere inside a still tank of water. What is pressing on it? Answer: the entire column of water sitting directly above it, all the way up to the surface. That water has mass, so gravity gives it weight, and that weight bears down on our point.

Go deeper and the column of water above you gets taller, so it weighs more, so it presses harder. That is the whole secret in a sentence: pressure in a liquid grows with depth, because there is more liquid stacked above pushing down. At the very surface there is nothing above you and the extra pressure is zero; a metre down, a metre's worth of water is on your shoulders; ten metres down, ten metres' worth.

The formula: p = \rho g h

We can turn "the weight of the column above" into an exact number. Picture a column of liquid of cross-sectional area A reaching down to depth h. Its volume is V = A h, so — using \rho = m/V from density — its mass is m = \rho V = \rho A h. Its weight is W = m g = \rho A h g. Now divide that force by the area it stands on, p = W/A, and the area A cancels clean away:

The area vanishing is the important part — it tells you the pressure at a given depth doesn't care how wide the column is. Each symbol has its own job:

• p — the pressure, in pascals (\text{Pa} = \text{N/m}^2);
• \rho — the liquid's density in \text{kg/m}^3 (water is 1000, the Greek letter "rho");
• g — the gravitational field strength, about 9.8\ \text{N/kg} on Earth;
• h — the depth below the surface, in metres.

Read it and the story leaps out: double the depth and you double the pressure; swap to a liquid twice as dense and you double the pressure again. Mercury, over thirteen times denser than water, pushes over thirteen times as hard at the very same depth.

Worked example 1 — pressure on a diver

A diver swims down to a depth of 12\ \text{m} in fresh water (\rho = 1000\ \text{kg/m}^3). Taking g = 9.8\ \text{N/kg}, what is the water pressure on her at that depth?

Step 1 — write the formula.

Step 2 — put the numbers in.

Step 3 — make sense of it. That is about 118\ \text{kPa} — roughly the same again as the atmosphere already pressing on us at the surface. So at 12 m the water alone is squeezing her about as hard as the whole sky does up top. Twice as deep, at 24 m, and it would double to 235{,}200\ \text{Pa}.

Worked example 2 — a denser liquid pushes harder

Take two identical tall jars and fill one with water (\rho = 1000\ \text{kg/m}^3) and one with mercury (\rho = 13{,}600\ \text{kg/m}^3). What is the pressure 0.50\ \text{m} below the surface of each?

Water:

Mercury:

Same depth, same g, same shape of jar — yet mercury pushes about 13.6 times harder, exactly the ratio of the two densities. Depth is only half the story; the liquid itself matters just as much. This is precisely how an old-fashioned barometer works: a column of mercury only 0.76\ \text{m} tall produces the same pressure as many kilometres of air.

Worked example 3 — working backwards to a depth

A research submarine's hull is rated to survive a water pressure of up to 4{,}900{,}000\ \text{Pa} (about 48 atmospheres). Treating the sea as having density 1000\ \text{kg/m}^3 and taking g = 9.8\ \text{N/kg}, how deep can it safely dive?

Here we know p, \rho and g and want h, so rearrange to h = \dfrac{p}{\rho g}:

Half a kilometre down and the sea is already squeezing with millions of pascals — which is why a deep-diving hull must be so fantastically strong. The units are a handy check: \text{Pa} \div (\text{kg/m}^3 \times \text{N/kg}) works out to metres.

Not the shape, not the amount — just the depth

Here is the fact that trips people up. Because the area cancelled out of p = \rho g h, the pressure at a given depth depends on nothing but the depth and the liquid. It does not depend on the shape of the container, the width of it, or how much liquid there is in total.

Fill a thin drinking straw and a wide bathtub each to a depth of 0.3\ \text{m} with water. The bathtub holds hundreds of times more water — yet at the bottom the pressure is exactly the same in both, 1000 \times 9.8 \times 0.3 = 2{,}940\ \text{Pa}. A tall, thin tube of water can push just as hard at its base as an ocean at the same depth. This surprising result is sometimes called the hydrostatic paradox: the pressure at the bottom is set by the height of the liquid, never by its volume.

Pressure pushes every way at once

You might think the liquid only pushes down — after all, it's the weight above that causes it. But a liquid is runny: it passes any squeeze on to its neighbours in all directions. The result is that at any point the pressure acts equally in every direction — up, down, sideways, and every angle in between.

That is why a submarine is squeezed inward from all sides at once, why water squirts sideways out of a hole in the side of a bucket, and why the pressure on your eardrum in the pool feels the same whether you tilt your head left or right. In the tank below, watch the arrows: they all point outward from the chosen point and they are all the same length, because the pressure there is the same in every direction — and every arrow grows as you drag the point deeper.

Drag Depth down and every arrow lengthens together — deeper means a bigger push, the same in all directions. Now switch the liquid from water to mercury: at the very same depth the arrows leap out to full length, because mercury is over thirteen times denser and so pushes over thirteen times harder.