How this instrument works
Sink beneath a liquid surface and what presses on you is the weight of everything stacked overhead. Picture a column of any cross-section running from your position up to open air: its mass is density times area times depth, its weight is that mass times gravity, and spreading such weight back over the same area cancels area completely. What survives is ρ·g·h, expressed in pascals, one pascal being one newton per square metre. That cancellation carries the whole idea — width, shape and total volume all drop out of it.
Simon Stevin set this down in 1586 in De Beghinselen des Waterwichts, arguing from imaginary rigid surfaces frozen inside a body of water. Blaise Pascal made it memorable around 1653 with a cask and a slender pipe: run a thin tube several metres above a sealed barrel, pour in barely a litre, and staves give way — because what governs is column height rather than that pipe's laughable weight. Textbooks still teach the demonstration as a hydrostatic paradox, and SI honours Pascal by naming its unit of pressure after him.
Three assumptions prop up this formula: density uniform through the column, gravity steady across that depth, liquid genuinely at rest. Gases break the first at once — air thins as it climbs, so atmospheric pressure follows an exponential barometric law rather than any straight line. Water is very nearly incompressible but not perfectly so, and kilometre-scale ocean columns want a measured in-situ density profile rather than one surface figure. Set liquid moving and Bernoulli's trade between speed and pressure takes over. What emerges here is gauge, reckoned against local atmosphere, not absolute reckoned from vacuum.
- Set Fluid density for whatever liquid you are under: 1000 kg/m³ fresh water, 1025 seawater, roughly 850 for diesel, 13595 for mercury. That menu also takes g/cm³ and lb/ft³.
- Enter Depth below the surface as straight-line vertical distance down from open air — never distance measured along a sloping tank wall or an angled pipe run.
- Read Gauge pressure at depth in pascals, or flip that field to kPa, bar, psi or atm. Pascals grow unwieldy quickly, so bar and psi are usually kinder to read.
- Wanting absolute pressure instead? Add your local atmospheric reading to whatever comes back — near sea level, 101325 Pa.
Worked example — ten metres of fresh water
A lake diver hovering 10 m down. Fluid density reads 1000 kg/m³, Depth below the surface reads 10 m, so P = 1000 × 9.80665 × 10 = 98066.5 Pa. Flip that output unit and identical arithmetic gives 98.07 kPa, 0.9807 bar, or 14.22 psi.
Look how near 98066.5 Pa sits to one standard atmosphere, 101325 Pa. Ten metres of water is very nearly one more atmosphere; 10.33 m is exactly one. Our diver therefore carries about twice sea-level loading in absolute terms, 98066.5 + 101325 = 199391.5 Pa. Steepest relative change happens in those first few metres, which is precisely why open-water instructors hammer on ear equalisation near the top rather than deeper down.
Questions
Does container shape or width change the reading?
Not at all. Only depth, density and gravity enter, because area cancels when a column's weight gets spread across its own base. A drinking straw and a swimming pool, sampled at matching depth, sit at matching pressure. Total force on a floor does differ, since force is pressure multiplied by area — a pool floor carries tonnes and a straw carries grams. Confusing those two quantities is the mistake people actually make here.
What separates gauge pressure from absolute pressure?
Zero point. Gauge counts upward from whatever atmosphere presses on your liquid surface; absolute counts from vacuum. This instrument returns gauge, so add 101325 Pa near sea level to convert. Tyre and boiler gauges read gauge too, which is why a flat tyre shows zero rather than one atmosphere. Diving tables and vapour-pressure work instead need absolute, and mixing them up shifts every answer by roughly one bar.
How deep is one atmosphere of water?
About 10.33 m in fresh water: 101325 ⁄ (1000 × 9.80665) = 10.332 m. Salt water, denser at roughly 1025 kg/m³, needs only 10.08 m for that same bar. Divers round to 10 m per atmosphere and accept a 3% error, which explains why 30 m of seawater is treated as 4 atmospheres absolute — three from liquid overhead, one from sky.
Why is blood pressure quoted in millimetres of mercury?
Because early sphygmomanometers really were mercury columns, and ρ·g·h converted column height straight into pressure. Mercury runs 13595.1 kg/m³, so 120 mmHg equals 0.120 × 13595.1 × 9.80665 ≈ 16.0 kPa. Torricelli's 1643 barometer used identical reasoning: 760 mm of that metal reproduces standard atmosphere, 101325 Pa. Clinical practice kept the unit long after glass columns left the ward, and hospital lung work still favours cmH₂O.
Can I use this formula for air or other gases?
Over short heights, yes. Air near sea level is about 1.2 kg/m³, so three metres of it contributes 1.2 × 9.80665 × 3 ≈ 35 Pa — real, measurable, and why a lift ride pops your ears. Over kilometres the answer degrades badly, since gas compresses under its own weight and density falls with altitude. Beyond a few hundred metres, reach for a barometric formula built on exponential decay instead.
How tall must a water tower be for decent mains pressure?
Height alone sets it. A tower holding water 40 m above your tap delivers 1000 × 9.80665 × 40 = 392266 Pa, near 3.9 bar or 57 psi — squarely inside the 40–80 psi band domestic plumbing expects. Storage volume changes nothing; a modest tank at 40 m beats a reservoir at 5 m. That is Stevin's paradox doing municipal work, and it is why towers are built tall rather than wide.