Hydrostatic Pressure Calculator

Yes, Hydrostatic Pressure Calculator is totally free :)

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Everything starts from P = ρ·g·h — the pressure a still fluid exerts at a depth h below its free surface — with the absolute pressure obtained by adding whatever pressure sits on that surface, normally one atmosphere. Pick the question you are actually asking: pressure at a depth, a pressure-versus-depth profile, the thrust and centre of pressure on a dam or gate, a U-tube or differential manometer, a hydraulic press, atmospheric pressure at altitude, a stack of immiscible layers, or the hydrostatic paradox. Every input is converted to SI, the whole calculation runs in SI, and rounding happens only when the numbers are printed. Each mode also recomputes its answer by a second, independent route and reports the residual, so you can watch the arithmetic check itself instead of trusting a single number.

Because P = ρ·g·h has no term for volume, width or shape — only the vertical distance down to the free surface appears in it. Fill a straight cylinder, an upward-narrowing cone and a flaring funnel to the same depth of 0.8 m with water and all three read exactly 7845.32 Pa at the base, even though the funnel holds seven times as much water as the cone. Multiply by a 0.02 m² base and every one of them pushes down with 156.9064 N. The cone only contains 52.30 N of water, so its base feels 104.60 N more than the fluid above it weighs; the sloping walls supply that difference by pushing down on the fluid. That mismatch between base force and fluid weight is the resolution of the paradox, not an error.

Not at mid-depth. The resultant acts at the centre of pressure, y_cp = ȳ + I_c/(ȳ·A), which always sits below the centroid because pressure grows with depth and the deeper strip of wall is pushed harder. For a wall reaching the free surface the formula collapses to the neat closed form y_cp = 2H/3, exactly H/6 below the centroid: a 6 m dam face 4 m wide carries 706078.8 N acting 4 m down, one third of the way up from the base. As a gate is sunk deeper the offset shrinks — a 2 m gate whose top edge is 3 m down has its centre of pressure only 8.3 cm below its centroid — because the pressure gradient becomes small next to the mean pressure.

Close, but not exact, and the tool quantifies the gap rather than repeating the rule of thumb. Ten metres of fresh water gives 1000 × 9.80665 × 10 = 98066.5 Pa, which is 0.9678 atm — about 3.2 % short. One full atmosphere of gauge pressure needs 10.3322745 m of fresh water, or 10.0803 m of seawater at 1025 kg/m³. That same number is why no suction pump can lift water more than about 10.33 m: a perfect vacuum above the column can only be balanced by 1 atm of head, and in practice the water boils (cavitates) before that, once the absolute pressure falls below its vapour pressure of roughly 2.3 kPa at 20 °C.

Because the two use different mercury. The millimetre-of-mercury unit is defined on mercury at 0 °C, 13595.1 kg/m³, while the mercury sitting in a laboratory manometer is at room temperature and closer to 13534 kg/m³. A 250 mm column computes to 33180.8003 Pa from ρgh at 13534, but an instrument with an mmHg scale reports 33330.5969 Pa — a real 0.451 % disagreement, not rounding noise. The calculator keeps the two densities strictly separate, offers 0 °C mercury as its own preset, and labels which density produced any mmHg figure, so the two paths are never silently mixed.

The arithmetic is exact to double precision and every mode is cross-checked by an independent second route, but the physics carries assumptions worth knowing. The fluid is treated as incompressible and of uniform density, so deep-ocean figures are underestimates — real seawater compresses several percent near 11 km, which the tool warns about. The fluid is assumed to be at rest, so nothing here covers flowing pipes, waves or accelerating tanks. Gravity is constant over the depth involved, and the barometric mode uses the standard-atmosphere lapse rate, which stops being valid above the tropopause at about 11 km. Gauge pressure is deliberately allowed to go negative, because suction and vacuum are perfectly physical.