Why only the depth matters
Take a horizontal slice of still fluid at depth h and ask what holds it up. The only thing available is the pressure difference across it, and the only thing pushing down is the weight of everything above. A column of cross-section A and height h holds a mass ρAh and therefore a weight ρAhg. Divide by the area and the A cancels: P = ρgh.
That cancellation is the whole result, and it produces the hydrostatic paradox. A narrow tube and a wide reservoir filled to the same height produce identical pressure at the bottom, even though one holds a hundred times more water. A dam is designed for the depth of the reservoir, not its surface area, and a 3 m garden pond exerts exactly the same pressure on its liner as a 3 m column in a drinking straw. Pascal demonstrated this in 1646 by bursting a strong barrel with a few cupfuls of water in a long thin pipe.
The same reasoning shows that pressure is the same everywhere at a given level in a connected body of fluid, which is why water finds its own level in a set of connected vessels, why a manometer works, and why a spirit level in tube form can transfer a datum around a building.
Gauge versus absolute matters enormously. Gauge pressure is measured relative to whatever is outside — normally the atmosphere — and is what an ordinary dial gauge, tyre gauge or diving depth gauge reads. Absolute pressure includes the atmosphere on top. At 10 m in seawater the gauge pressure is about 100.5 kPa but the absolute pressure is about 202 kPa, which is why a diver's lungs experience roughly two atmospheres and why gas laws must always be applied in absolute terms.
Head, and why engineers measure pressure in metres
Rearranged, h = P/(ρg), and that height is called the head. It is a genuinely useful way to express pressure, because pumps deliver a head that is almost independent of the fluid's density: a centrifugal pump that lifts water 30 m will also lift a light hydrocarbon roughly 30 m, though the pressure rise in pascals will be lower and the power required lower with it.
Two conventional head units appear constantly. One metre of water column is defined as 9,806.65 Pa — that is 1,000 kg/m³ at standard gravity, a definition rather than a measurement, so it does not drift with the temperature of your actual water. One millimetre of mercury (torr) is 133.322 Pa, from 13,595.1 kg/m³ at 0 °C. Those two definitions give the classic values: standard atmosphere = 760 mmHg = 10.332 m of water = 33.90 ft of water = 14.6959 psi = 101.325 kPa.
The water figure has a hard practical consequence. A suction pump cannot lift water more than 10.33 m even in principle, because it works by removing pressure above the water and letting the atmosphere push it up — and the atmosphere has only 101.325 kPa to give. In practice cavitation, vapour pressure and friction reduce the workable limit to about 7–8 m, which is why deep wells use submersible or jet pumps rather than surface suction. Net positive suction head (NPSH) is this calculation applied at the pump inlet.
For a tank of unusual liquid, the shortcut is specific gravity. A liquid of SG 0.87 produces 87% of the pressure of water at the same depth, so 10 m of crude oil is 85.3 kPa rather than 98.1 kPa. That is why a tank gauge calibrated in metres of product must be corrected when the product changes.
Worked example: a rectangular water tank and the force on its wall
A tank holds fresh water at 20 °C (ρ = 998.2 kg/m³) to a depth of 4.50 m. The tank is open to the atmosphere. One wall is 3.00 m wide. What is the pressure at the bottom, and what total force does the water exert on that wall?
- Pressure gradient.
ρg= 998.2 × 9.80665 = 9,789.00 Pa per metre of depth. - Gauge pressure at the bottom. P = 9,789.00 × 4.50 = 44,050.5 Pa = 44.05 kPa, which is 6.389 psi.
- Absolute pressure at the bottom. 44,050.5 + 101,325 = 145,375.5 Pa = 145.38 kPa, or 1.4347 atm.
- Equivalent head. 44,050.5 ÷ 9,806.65 = 4.492 m of water column — slightly less than 4.50 m because the conventional metre of water uses 1,000 kg/m³ while this water is at 998.2.
- Force on the wall. Pressure rises linearly from zero at the surface to 44,050.5 Pa at the bottom, so the average over the wall is half the maximum: 22,025.3 Pa. Wall area = 4.50 × 3.00 = 13.5 m². Force = 22,025.25 × 13.5 = 297,341 N ≈ 297 kN, about 30 tonnes-force.
- Where it acts. Because the pressure distribution is a triangle, its centroid lies one third of the way up from the bottom, at 1.50 m. That is the point a structural engineer applies the resultant to when checking the wall's overturning moment.
Notice the two things the calculation did not need: the tank's length, and how much water it holds. Change the tank from 3 m long to 30 m long and the pressure at the bottom is identical. Only the force on the larger wall changes, and only because the area did.
Reading the result and knowing its limits
Check the gradient first. Fresh water gives close to 9.81 kPa per metre, or 0.433 psi per foot; seawater gives 10.05 kPa per metre. If your answer implies a gradient far from those, either the density or a unit is wrong. Mercury is the useful outlier at 133.3 kPa per metre — 13.6 times water, which is exactly what makes a mercury barometer 760 mm tall instead of 10.3 m.
Then decide whether you need gauge or absolute. Structural checks, pipe ratings and pump duties are almost always gauge, because the atmosphere acts on both sides of the wall and cancels. Anything involving gas — a diver's breathing gas, a bubble rising, a vapour-pressure or cavitation check, or the ideal gas law — must use absolute pressure, or the answer will be wrong by one atmosphere.
The constant-density assumption is excellent for liquids over ordinary depths and poor in two places. Water is not perfectly incompressible: over several kilometres of ocean the density rises measurably, so integrating the true density profile gives a pressure a few percent above the constant-density ρgh estimate. And for gases, density falls with height, so atmospheric pressure follows an exponential barometric formula rather than a straight line; P = ρgh is only usable for a gas over a few metres.
Two other qualifications matter in the field. This is a static result: as soon as the fluid moves, Bernoulli's equation redistributes pressure into velocity, and friction losses subtract from it along a pipe. And the depth must be measured vertically. A 100 m pipeline running along a level field has no hydrostatic component at all; the same 100 m of pipe dropped straight down a shaft has 981 kPa of it.
Pressure at depth in fresh water and seawater
| Depth | Fresh water gauge (kPa) | Seawater gauge (kPa) | Seawater gauge (psi) | Seawater absolute (atm) |
|---|---|---|---|---|
| 0 m | 0.00 | 0.00 | 0.00 | 1.000 |
| 5 m | 48.95 | 50.26 | 7.29 | 1.496 |
| 10 m | 97.89 | 100.52 | 14.58 | 1.992 |
| 20 m | 195.78 | 201.04 | 29.16 | 2.984 |
| 30 m | 293.67 | 301.55 | 43.74 | 3.976 |
| 40 m | 391.56 | 402.07 | 58.32 | 4.968 |
| 50 m | 489.45 | 502.59 | 72.89 | 5.960 |
| 100 m | 978.90 | 1005.18 | 145.79 | 10.920 |
The diving rule that every 10 m of seawater adds one atmosphere is accurate to about 0.8%; the exact figure is 0.992 atm per 10 m. Deep values ignore water compressibility, which becomes significant below a few kilometres.
Mistakes that give the wrong pressure
- Using pipe length instead of vertical drop. Only the vertical component counts. A sloping run contributes nothing beyond the height it descends.
- Mixing gauge and absolute. Adding the atmosphere twice, or forgetting it in a gas-law calculation, is the single most common error. Gas volumes and bubble sizes need absolute pressure.
- Assuming the container's shape matters. It does not. A wide tank and a narrow tube of the same depth give identical bottom pressure.
- Forgetting temperature and salinity in the density. Seawater at 1,025 kg/m³ produces 2.7% more pressure than fresh water; hot water produces slightly less than cold.
- Using a metre-of-water figure calibrated on the wrong density. The conventional metre of water is fixed at 1,000 kg/m³, so it is not identical to a metre of your actual 998.2 kg/m³ water.
- Applying ρgh to a gas column. Gas density falls with height, so the relation only works over a few metres before the exponential barometric formula is needed.
- Ignoring compressibility at extreme depth. Below a few kilometres the constant-density result understates the true pressure by a few percent.
- Expecting a suction pump to lift water more than 10.33 m. That is the whole atmosphere's worth; practical limits are lower still because of vapour pressure and friction.
Where hydrostatics leads next
Buoyancy follows immediately. Because pressure grows with depth, the upward push on the bottom of a submerged body exceeds the downward push on its top, and the difference is exactly the weight of the displaced fluid — Archimedes' principle, which is a corollary of P = ρgh rather than a separate law. The same pressure difference explains why a ship floats and why a submarine's hull must resist a crushing net load.
Hydraulics is the other direct descendant. Pascal's principle says a pressure applied to an enclosed fluid is transmitted undiminished everywhere, so a small force on a small piston becomes a large force on a large one in proportion to the area ratio. Car brakes, jacks and excavator rams are all built on it, and the working pressures involved dwarf anything hydrostatic: 20 MPa in a hydraulic circuit is the equivalent of 2 km of water head.
Once fluid starts moving, hydrostatics becomes only one term in the energy balance. Bernoulli's equation trades pressure head for velocity head and elevation head, and friction losses depend on whether flow is smooth or turbulent — a distinction set by the Reynolds number. For a gas rather than a liquid, the density that appears in every one of these expressions comes from the ideal gas law, which is also what tells a diver how a lungful of air at 30 m expands on the way up. And if you are heating or cooling the fluid, the density change comes from the same thermal data used in a heat energy calculation.
Diving and pressure vessels carry real risk
The gas volume in a diver's lungs varies inversely with absolute pressure, so ascending from 10 m while holding your breath doubles it. Use this calculator for physics, and certified dive tables, a dive computer and proper training for anything in the water. Pressure vessel design is governed by codes such as ASME BPVC, not by ρgh alone.