Hydrostatic pressure:
what the water pushes with at a given depth

Pressure as the weight of the liquid above

Pick any point inside still water. Directly above it is a column of liquid reaching the surface, and the weight of that column presses down on whatever sits at the bottom. Divide that weight by the area of the column's base and you have the pressure.

This is why only depth matters. A narrow pipe and a wide lake give the same pressure at the same depth, as long as the liquid is the same. The width of the container cancels out of the sum, an idea sometimes called the hydrostatic paradox.

The pressure also acts equally in every direction at that point, so the same figure presses sideways on a wall and upward on a ceiling, which is why submarines are built as rounded hulls.

Deriving the formula from weight and area

A column of base area A and height h holds volume A × h. Its mass is density × A × h, and its weight is that mass times g. Divide by A and the area drops out, leaving density, g and depth.

Both versions need density in kilograms per cubic metre. If you have it in g/cm³, multiply by 1000; water at 1 g/cm³ is therefore 1000 kg/m³.

Worked example: three metres of fresh water

A sensor sits 3.0 m below the surface of a freshwater tank. Find the pressure caused by the water alone, and then the total pressure on the sensor.

The gauge value is the number a depth transducer usually reports, because it is measured relative to the air above the water. The absolute value is the one you need for gas calculations, such as how much a bubble would compress at that depth.

The pattern scales linearly: 6 m gives 58.86 kPa gauge and 10 m gives 98.1 kPa, just under one atmosphere. The rule of thumb that every 10 m of water adds about one atmosphere follows from that last figure.

How different liquids compare

Seawater is about 2.5 percent denser than fresh water, so it gives a slightly higher number at the same depth. Mercury's density explains why barometers are only about 0.76 m tall, whereas a water barometer would need roughly 10.3 m.

Densities shift a little with temperature and salinity, so treat tabulated values as typical and look up the figure for your actual liquid when precision matters.

Limits of the simple depth formula

• Moving liquids. Once the fluid flows, dynamic effects change the pressure, and Bernoulli's equation applies instead.
• Very deep water. Density rises slightly with depth because water compresses, which matters at several kilometres, so a constant density becomes an approximation.
• Layered liquids. Add each layer separately, using its own density and thickness.
• Non-vertical depth. Use vertical height only; a long sloping pipe does not add pressure beyond its vertical rise.
• Accelerating containers. In a lift or a vehicle the effective g changes, and so does the pressure.

Dams, towers and water supply

Because pressure rises in step with depth, structures that hold water must be strongest at the bottom. At 20 m depth in fresh water the gauge pressure is 1000 × 9.81 × 20 = 196,200 Pa, about 196 kPa, nearly twice atmospheric pressure.

Pressure on a vertical wall is not uniform; it grows from zero at the surface to its maximum at the base. The average over the wall is half the bottom value, so the total force per metre of wall width is ½ × ρ × g × h². For a 20 m wall that is 0.5 × 1000 × 9.81 × 400 = 1,962,000 N, close to 2 meganewtons on every metre of width.

The same relation explains water towers. A tank 30 m above a tap gives about 1000 × 9.81 × 30 = 294,300 Pa of static pressure there, roughly 2.9 bar, before pipe friction and flow losses are considered. Raising the tank by 10 m adds roughly one bar.

Sanity checks for a depth-pressure result

• Each metre of fresh water should add close to 9.8 kPa. If your answer for 3 m is nowhere near 29 kPa, check the units for density.
• Pressure should rise in a straight line with depth. Doubling the depth must double the gauge reading.
• Seawater should always give a slightly higher answer than fresh water at the same depth.
• A sealed gauge on a tank may show gauge pressure; one connected to a vacuum line may show absolute pressure.

A diver's depth gauge converts exactly this relation: the sensor reads absolute pressure, subtracts the surface value and divides by ρ × g to display metres. That is why gauges calibrated for fresh water read slightly shallow when used in the sea.

Using depth and pressure interchangeably

Because pressure is directly proportional to depth, you can run the formula backwards to find a depth from a pressure reading. Divide the gauge pressure by density times g. A gauge reading of 49,050 Pa in fresh water points to 49,050 ÷ 9,810 = 5 m of water above the sensor.

Water engineers sometimes quote pressure as 'head', meaning the height of a water column. 10 m of head is about 98 kPa. Pumps and tanks are specified in head because it does not depend on the liquid's density once you stay with water.

If a pressure in atmospheres or bar needs converting, a unit converter such as an atmosphere-to-pascal tool handles the 101,325 Pa conversion cleanly before you add or subtract.