Hydrostatic Pressure Calculator
Pressure underwater is just the weight of the water above, divided by the area it leans on. Enter the fluid’s density and your depth, and this calculator returns the pressure — gauge (above atmospheric) and absolute — the number divers, dam engineers and ear drums all live by.
Weight per Area: The Whole Derivation
Stand a column of fluid on a surface: its weight is density × volume × g = ρ × (area × depth) × g, and pressure is that weight divided by the area — the area cancels, leaving P = ρgh, the most economical formula in fluid mechanics. Depth and density only: not the container's shape, not the total volume, not the distance to the far shore — the hydrostatic paradox that Stevin demonstrated in the 1580s and that still ambushes intuition (a dam holding back an ocean and a dam holding back a narrow fjord of the same depth feel the same pressure at their bases, metre for metre — the ocean's extra water presses on the fjord's other shores, not on this dam). The calculator's two outputs separate the fluid's contribution (gauge) from the atmosphere's ever-present one piled on top (absolute), because different professions pay rent to different sums.
Worked Example: Ten Metres of Fresh Water
The defaults price the first rung of the diver's ladder: ρ = 1,000, h = 10 m. Gauge pressure = 1,000 × 9.81 × 10 = 98,100 Pa (0.981 bar) — within 3% of a full atmosphere, which is the field rule divers actually carry: an atmosphere per ten metres. Absolute pressure there is 98,100 + 101,325 = 199,425 Pa ≈ 1.97 atm: the air in the diver's lungs is compressed to half its surface volume (Boyle's law collecting the consequence), the regulator delivers gas at ambient pressure, and a tankful lasts half as long as at the surface — this page's single multiplication silently governing dive planning, submarine hull ratings and the depth at which a snorkel becomes a straw in a vice.
The Paradox's Working Career
Pascal's barrel is the paradox as theatre: a long thin tube of water inserted into a sealed barrel bursts the barrel, because the tube's height (its depth term) generates pressure its few litres could never suggest by weight — ρgh with a carnival h. The same indifference to volume runs the hydraulic press (a small force on a small piston, transmitted by pressure to a large piston, multiplies by the area ratio — the hub's lever logic in fluid dress), the artesian well, and the barometer: mercury's 13,546 density in this calculator returns the atmosphere's height in that fluid as 0.76 m, which is why barometers are mercury-sized and water barometers would need a 10.3 m stairwell. Enter the densities and depths of your own apparatus; the formula has never once asked how much fluid you own.
The Water Tower's Quiet Logic
A water tower is this formula as municipal policy: raise the tank 30 m and every tap below it receives ρgh ≈ 294 kPa — about 3 bar — of pressure with no pump running, the elevation doing the work gravity is paid nothing for. Cities are plumbed in contours of this equation; a house on a hill above the tower's base lives in a different pressure world from the valley floor's, and pressure-reducing valves exist to forgive the difference. The calculator's depth field does not care whether the "depth" is ocean over a diver or tank above a tap — which is the surest sign the formula has hold of something real: the same multiplication, at every scale water is stored or swum in.
The Ear's Complaint
Every swimmer has felt this formula personally: descend two metres in a pool and the gauge pressure — 1,000 × 9.81 × 2 ≈ 19.6 kPa of it — squeezes the eardrum inward, and the equalising pinch of the nose is the body renegotiating the pressure this page computes. The shallowness of the injury threshold is the lesson: the eardrum notices kilopascals, the formula's smallest change of depth, which is why freedivers train equalisation before depth and why aircraft cabins (the same physics, air's small density, thousands of metres) are pressure-managed as a medical system, not a comfort feature. The calculator's numbers are gentle abstractions until they are pressing on a membrane you own.
Communicating Vessels: Stevin's Corollary
Connected containers of a fluid settle to one level, because any height difference is a pressure difference the fluid immediately spends on flowing — the spirit level, the water gauge on a boiler, the lock-keeper's pound and the coffee-pot spout all run on the equalisation. Surveyors levelled aqueducts across Roman provinces with water-filled trenches obeying it. The calculator's depth is measured from that common level wherever your apparatus finds it — which is why the formula asks for height below the surface, and why the surface's flatness, guaranteed by the law itself, is the reference every measurement inherits. It is a pleasingly circular foundation: the law that prices the pressure also certifies the ruler the pressure is priced with.
Related Tools
More from the Physics Calculators hub: the Pressure Calculator for force-over-area in the dry world, and the Stokes' Law Calculator — the other great ρ-driven account of what fluids do to small spheres.
Frequently Asked Questions
Why gauge and absolute — who uses which?
Divers, tyre gauges and most pressure instruments read gauge (zeroed against the local atmosphere, because that is what their diaphragms feel), while physics, chemistry and altitude work need absolute (zeroed against vacuum, because gas laws and vapour pressures answer to the total). The difference is always exactly one atmosphere — 101,325 Pa at sea level standard — and the confusion between the two has popped more than one student problem and at least a few industrial vessels. The calculator prints both and labels them, which is the whole professional discipline: never quote a pressure without declaring its zero.
How deep before the pressure is dangerous to a human?
The pressure itself is not the injury — a body is mostly water and transmits it — the dangers are its consequences: lungs must be supplied gas at ambient pressure (below a few metres, no human can inflate their lungs against the squeeze through a surface snorkel: at 1 m the extra 9.8 kPa already defeats most inspiratory muscles' comfort), dissolved nitrogen accumulates with pressure × time (the bends on ascent), and oxygen turns toxic at partial pressures recreational divers approach near 40–60 m. This page supplies the pressure term of every one of those calculations; dive medicine multiplies it by time and physiology.
Does the formula work in air — for a building's height?
Yes, with air's tiny density (about 1.2 kg/m³): pressure falls by roughly 12 Pa per metre near sea level — the stack effect that drives a tall building's winter drafts (warm indoor air, less dense, makes the effect asymmetric floor by floor) and the calibration of barometric altimeters. Air's compressibility bends the simple linear rule over kilometres (density itself falls with height — the exponential atmosphere), but for a building, a hill or a mine shaft the linear ρgh with local density is the working figure, and the calculator's density field is where the honesty goes in.
What pressure is at the bottom of the Mariana Trench?
Depth about 10,900 m, seawater 1,025: P = 1,025 × 9.81 × 10,900 ≈ 109.6 million Pa — about 1,080 atmospheres, a tonne of weight on every square centimetre. The calculator returns it from the same two inputs as the paddling-pool case (the formula does not know the trench is special), and the number explains the bathyscaphe Trieste's spherical cabin, the implosion hazards of lost submersibles, and the strangeness of hadal biology, whose cell membranes are engineered against a pressure that would crush a surface animal's biochemistry into different shapes.
Why does a dam get thicker toward the bottom?
Because this page's pressure grows linearly with depth, the total force on the dam wall grows with depth squared (pressure integrated over the wall's height), and the overturning moment with depth cubed — so the structure's answer is a triangular profile, the gravity dam's cross-section being nearly a portrait of the ρgh line itself. Hoover Dam's base is 201 m thick against a 221 m height: read the profile and you are reading this formula in concrete. Small garden-pond walls obey the same drafting logic at centimetre scale, which is why even they flare at the foot.
Does it matter that the fluid is moving?
The law is named hydrostatic for a reason: it is exact for still fluids, and a very good approximation for slowly moving ones. Once the fluid flows, pressure also trades against speed (Bernoulli's exchange — faster flow, lower pressure), and the depth rule becomes one term of a fuller accountancy rather than the whole of it. A river's bed still feels close to ρgh from the water above it; a fire hose's jet does not. Knowing which regime you are in is the professional judgement the formula's simplicity tempts you to skip — this page computes the still-water answer, and says so on its face.
Who first stated the law?
Simon Stevin, in 1586, worked out the depth rule and the paradox (publishing in Dutch, for engineers); Pascal gave it its theatrical proofs and its unit-hungry name a lifetime later; Torricelli's mercury column (1643) made the atmosphere's own pressure undeniable in between. The formula's triple paternity is typical of fluid statics — it is simple enough to be discovered by anyone who thinks clearly about a column of water, which is also why it remains the first page of every fluids course and, now, of this hub's treatment of pressure at depth.