Pressure and Hydrostatic Pressure Calculator: P = F/A and ρgh
Pressure and Hydrostatic Pressure Calculator
Two pressures on one page, and the question is always which one the examiner wants. P = F/A for a force spread over a contact area, and P = P₀ + ρgh for a point inside a liquid — and a switch that lets you answer in gauge pressure or in absolute pressure, which is the distinction that costs the marks.
Contact pressure, hydrostatic pressure, gauge or absolute
A diver is 2.00 m below the surface of fresh water (ρ = 1000 kg/m³). Take g = 9.81 m/s² and the air above the water at one standard atmosphere. Find the absolute pressure at that depth. Separately, a 51 kg crate presses down with 500 N over a base of 0.25 m² — find the pressure under it.
Two formulas, two situations, and the switch between gauge and absolute
- F
- the force pressing perpendicular to the surface, in newtons. Only the component at right angles to the surface counts.
- A
- the contact area in square metres. Halve the area and you double the pressure for the same force, which is the whole idea behind a drawing pin and a snowshoe.
- ρ
- density of the liquid in kg/m³. Constant, in this model.
- h
- vertical depth below the free surface, in metres.
- P₀
- pressure acting on the free surface. Set it to zero and you get gauge pressure; set it to 101325 Pa and you get absolute pressure.
Worked example
A diver is 2.00 m below the surface of fresh water (ρ = 1000 kg/m³). Take g = 9.81 m/s² and the air above the water at one standard atmosphere. Find the absolute pressure at that depth. Separately, a 51 kg crate presses down with 500 N over a base of 0.25 m² — find the pressure under it.
Gauge pressure first, because that is the part the water contributes. ρgh = 1000 × 9.81 × 2.00 = 19 620 Pa, which is 19.62 kPa. Notice that nothing about the pool's size appeared — a 2 m deep bath and a 2 m deep lake give the identical figure.
Add the surface pressure to get absolute pressure. P = P₀ + ρgh = 101 325 + 19 620 = 120 945 Pa = 120.945 kPa. That is 1.1936 atmospheres, so being 2 m under water adds about a fifth of an atmosphere.
Sanity-check the depth that adds one atmosphere. 101 325 ÷ (1000 × 9.81) = 10.33 m of water. The familiar diving rule that you gain roughly one atmosphere every 10 metres falls straight out of ρgh, and it is worth remembering as a check on any answer of this kind.
Now the completely separate contact-pressure part. P = F/A = 500 ÷ 0.25 = 2000 Pa = 2.00 kPa. There is no ρ, no g and no depth in this one. Mixing the two formulas — putting a depth into F/A, or an area into ρgh — is the commonest wrong turn on this topic.
Compare them, which is the useful bit. 2000 Pa of crate pressure is the same as being only 0.204 m — about 20 cm — under water. A crate that feels heavy standing on your foot is doing less to you than a shallow paddling pool would.
Where the marks go. Depth in centimetres fed in as metres is a factor of 100. Area in cm² fed in as m² is a factor of 10 000. And quoting 120.945 kPa when the question asked for gauge pressure, or 19.62 kPa when it asked for absolute, loses the mark even though the arithmetic was perfect.
The same pressure written six ways
| Pressure | Pa | kPa | atm | bar | mmHg | Depth of water giving it |
|---|---|---|---|---|---|---|
| Gauge pressure 2 m under water | 19 620 | 19.62 | 0.1936 | 0.19620 | 147.2 | 2.00 m |
| One standard atmosphere | 101 325 | 101.325 | 1.0000 | 1.01325 | 760.0 | 10.33 m |
| This page’s default answer | 120 945 | 120.945 | 1.1936 | 1.20945 | 907.2 | 12.33 m |
| A car tyre, 32 psi gauge | 220 600 | 220.6 | 2.177 | 2.206 | 1655 | 22.49 m |
| The crate’s contact pressure | 2 000 | 2.000 | 0.01974 | 0.02000 | 15.00 | 0.204 m |
| Deepest point of the ocean, roughly | 110 000 000 | 110 000 | 1086 | 1100 | 825 100 | 11 200 m |
Gauge or absolute: which does each instrument give you?
| Instrument or quantity | Reads | So to get the other one |
|---|---|---|
| Tyre pressure gauge | Gauge | Add one atmosphere |
| Blood-pressure cuff (mmHg) | Gauge | Add 760 mmHg |
| U-tube manometer, one limb open | Gauge | Add the atmospheric reading |
| Barometer | Absolute | It is the atmospheric pressure |
| “Pressure at the bottom of the tank” in a typical exam question | Depends on the wording | Read it twice: “due to the liquid” means gauge |
| A tank sealed and pressurised to 3 bar | Usually gauge | Absolute is 4.01325 bar |
Why only the depth matters, and why gauge pressure is the one that fools people
The whole of hydrostatics sits on one sentence: the pressure at a point in a liquid at rest depends on how far down it is and nothing else. Not on the volume above it, not on the width of the vessel, not on whether the sides slope in or out. A swimming pool and a drinking straw, both filled to 2 m, have the same pressure at the bottom. This feels wrong, and the reason it is right is that the sloping walls of a wide vessel carry part of the weight — the liquid does not have to.
That result, sometimes called the hydrostatic paradox, is worth more than the formula. It tells you that a dam is designed around its depth rather than the size of the reservoir behind it, and that the pressure in a town’s water mains is set by the height of the service reservoir on the hill, not by how much water is in it. If a question gives you a volume or a mass of liquid and asks for the pressure at a point, you need the depth and the density; the volume is there to see whether you know that.
The distinction that actually loses marks is gauge against absolute. Every instrument with a dial on it — tyre gauge, cuff, manometer with one limb open to the air — compares the pressure inside with the air outside and reports the difference. That is gauge pressure. Absolute pressure is measured from a vacuum, and is the gauge reading plus the atmospheric pressure. The two differ by about 101 kPa, which is a large number in a school problem: a tank at “2 bar” is at 2 bar gauge and 3 bar absolute, and only one of those goes into the gas laws. Any formula derived from the kinetic theory of gases — the ideal gas equation among them — needs absolute pressure, which is why the switch on this page exists.
The second idea on the page, P = F/A, is a different thing wearing the same unit. It is about spreading a force out or concentrating it. The same 500 N under a crate’s 0.25 m² base gives 2 kPa; under the point of a drawing pin, perhaps 0.1 mm² of contact, it gives 5 billion pascals, which is why one goes into the wall and the other does not. Nothing about depth or density appears, and nothing about force or area appears in ρgh. Keeping them apart is the skill.
Where this goes wrong is almost always a unit. Area in square centimetres fed into a formula expecting square metres is out by a factor of 10 000, and the answer is wrong by four orders of magnitude without looking absurd — 2000 Pa and 20 000 000 Pa are both just numbers. Depth in centimetres costs a factor of 100. Density in g/cm³ instead of kg/m³ costs a factor of 1000: water is 1 g/cm³ and 1000 kg/m³, the same substance written two ways. Convert first, write the unit beside every number, and the arithmetic looks after itself. The significant figures page covers how many digits to keep once you have it.
Two things this model ignores, and both are fine at school level. Liquids are treated as incompressible, so ρ stays constant with depth; in the deep ocean the density actually rises by a few per cent and the real pressure at 11 km is slightly above the ρgh figure. And g is treated as constant, which it is to better than a part in a million over any depth you will meet. Neither changes a school answer. What does change the answer is the surface pressure, and that is a choice the question makes for you rather than a physical uncertainty.
One last habit worth building: convert every pressure into a depth of water as a sanity check. One atmosphere is 10.33 m. A tyre at 32 psi gauge is 22.5 m of water. Blood pressure of 120 mmHg is 1.63 m of water, which is roughly the height of the column of blood a standing person’s heart has to support — which is not a coincidence, and is why blood pressure is still quoted in millimetres of mercury rather than in pascals.
Frequently asked questions
Does the shape of the container change the pressure at the bottom?
No. Only the vertical depth below the free surface and the density of the liquid enter ρgh. A narrow tube, a wide tank and a vessel that flares outwards, all filled to the same level, have the same pressure on the base. The total force on the base is different, because force is pressure times area, but the pressure is identical. If you find that implausible, that is the right reaction — it is called the hydrostatic paradox for a reason, and the resolution is that the slanting walls push on the liquid too.
Should I add atmospheric pressure or not?
Read the wording. “Pressure due to the liquid”, “excess pressure” and “gauge pressure” all mean ρgh alone. “Total pressure” and “absolute pressure” mean you add P₀. If you are about to feed the number into a gas law, you need absolute pressure whatever the question called it, because the gas laws are written in absolute terms.
Why is blood pressure measured in millimetres of mercury?
Because the original instrument was a mercury column, and 120 mmHg meant the cuff pressure supported a 120 mm column. The unit stuck. One conventional millimetre of mercury is 133.322 387 415 Pa, which is just ρgh for mercury at 13 595.1 kg/m³ and standard gravity, computed once and then fixed by agreement so the unit does not drift with the room temperature. 120 mmHg is 16.0 kPa, gauge.
Is pressure a vector? It seems to have a direction.
Pressure is a scalar. The force it produces on a surface has a direction — always perpendicular to that surface — but the pressure at a point has no direction of its own, which is why the diagram above shows arrows pointing out in every direction from the same point. That is also why pressure adds as a plain number: P₀ + ρgh, with no components and no cosines.
Why does one atmosphere come out as 10.33 m of water?
Set ρgh equal to 101 325 Pa and solve for h: 101 325 ÷ (1000 × 9.81) = 10.33 m. It is the same calculation as the 760 mm mercury barometer, with the density 13.6 times smaller and the height therefore 13.6 times greater. It is the reason a suction pump cannot lift water from a well deeper than about ten metres however good the pump is.
The calculator gives a gauge pressure of zero when I set the depth to zero. Is that a bug?
No. At the free surface there is no liquid above you, so the liquid contributes nothing and the pressure is whatever is pushing on the surface. With P₀ set to one atmosphere the headline is then 101.325 kPa, and with P₀ at zero it is genuinely 0 kPa gauge. Both are correct answers to the question as set.
Related calculators
References
- NCERT, Physics: Textbook for Class XI, Part II, Chapter 9 “Mechanical Properties of Fluids”, §9.2. Equation 9.1 defines average pressure as F/A; Equation 9.6 gives P2 − P1 = ρgh; Equation 9.7 gives P = Pa + ρgh; Equation 9.8 is the mercury barometer. Read through IIT Kanpur’s SATHEE mirror of the chapter, because ncert.nic.in refuses the file to an automated request (robots).
- BIPM, Resolution 4 of the 10th CGPM (1954), “Definition of the standard atmosphere”: “1 standard atmosphere = 1 013 250 dynes per square centimetre, i.e. 101 325 newtons per square metre”. The resolution adopts the definition for general use, and does not itself use the word pascal.
- BIPM, The International System of Units (SI), 9th edition (2019, updated 2026). The newton, the pascal and the kilogram: the kilogram is a unit of mass and the newton a unit of force, and the brochure keeps them in separate tables for the reason this page keeps repeating.
- NIST, CODATA Internationally Recommended 2022 Values of the Fundamental Physical Constants, ASCII table read 11 November 2026. Standard acceleration of gravity 9.806 65 m s−2, marked (exact) rather than carrying an uncertainty; Newtonian constant of gravitation 6.674 30(15)×10−11 m3 kg−1 s−2.
physics.nist.gov/cuu/Constants/Table/allascii.txt