Pressure calculator for solids and fluids

Pressure is force divided by the area it acts on: P = F ÷ A, measured in pascals (Pa), where 1 Pa is 1 newton per square metre. A 1,200 N crate on a 0.4 m² base presses on the floor with 1,200 ÷ 0.4 = 3,000 Pa. In a liquid or gas, the pressure from the fluid at depth h is P = ρgh. This pressure calculator does both, rearranges either formula for any value, and gives the answer in kPa, bar, atm, mmHg and psi as well.

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Pressure Calculator

Calculate pressure in solids and fluids, including upthrust

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What the pressure calculator does

The calculator has three topics, chosen along the top row of Calculate:

• P = F/A: pressure on a surface, from a force and an area. It also finds the force from the pressure and area, or the area from the force and pressure.
• Fluids: pressure at a depth in a liquid or gas with P = ρgh. It also finds the depth from a pressure, or the fluid's density from a pressure and a depth.
• Upthrust: the buoyant force on an object in a fluid, U = ρVg, and whether the object would rise or sink. The buoyancy calculator guide covers it.

Every answer comes with the formula, the numbers put in and the result, and a graph or diagram. For P = F/A it draws pressure against area for your force, with your area marked. For a fluid it draws a column of the fluid with pressure arrows that grow with depth.

How to use the pressure calculator

1. Open the Pressure Calculator. Under Calculate, pick the topic (P = F/A, Fluids or Upthrust), then what you want to find.
2. Pick your units under the inputs. Force can be N, kN or lbf; area mm², cm², m² or ft²; pressure Pa, kPa, MPa, bar, atm, mmHg or psi; and depth mm, cm, m or km, among others.
3. Type the values you know. For a fluid, Common values fills in the density of air, petrol (gasoline in the US), olive oil, water, seawater or mercury.
4. Leave Gravity blank to use 9.81 m/s², or pick another planet from its list.
5. Read the answer and the working. The answer is in SI units (Pa, N, m², m or kg/m³), and Also equal to lists it in the other units.
6. To practise, press Random problem under Practice and help.

The pressure formula, P = F/A

Rearranged, F = PA and A = F ÷ P. The same force spread over a bigger area gives a smaller pressure, which is why snowshoes stop you sinking and a sharp knife cuts: its edge has a tiny area. On a graph of pressure against area for a fixed force, the curve falls steeply at first: halve the area and the pressure doubles.

Here x is the area in square metres and y is the pressure in pascals. Doubling the area from 0.4 m² to 0.8 m² halves the pressure from 3,000 Pa to 1,500 Pa.

Fluid pressure: P = ρgh

The deeper you go, the more fluid sits above you, so the pressure rises in a straight line with depth. It doesn't depend on the shape of the container, only on the depth, the density and g. In water, each metre adds 1,000 × 9.81 × 1 = 9,810 Pa, about 9.8 kPa.

Here x is the depth in metres and y is the pressure from the water in kilopascals. At 10 m it is 98.1 kPa, almost one atmosphere.

Gauge pressure and absolute pressure

ρgh is only the pressure from the fluid. The air above the surface pushes down too, so the total, or absolute pressure, is ρgh plus atmospheric pressure, 101,325 Pa. A pressure measured above atmospheric, like ρgh or a tyre (tire) gauge reading, is called gauge pressure. The calculator shows both: the answer is the fluid's pressure, and With the atmosphere on top adds 101,325 Pa. The In atmospheres reading is the fluid's pressure alone, in atm.

Pressure units

Worked examples you can check in the calculator

Example 1: pressure under a crate

A crate weighing 1,200 N sits on a base of 0.4 m². Choose P = F/A and Pressure, then type F = 1200 and A = 0.4.

That is 3 kPa, or 0.435 psi in the Also equal to list.

Example 2: a heel and a flat shoe, in cm²

A person weighing 600 N puts all their weight on one stiletto heel with an area of 1 cm². Set the area unit to cm² and type A = 1. The calculator converts 1 cm² to 0.0001 m² and gives P = 600 ÷ 0.0001 = 6 × 10⁶ Pa, which is 6 MPa or about 870 psi. Standing on two flat shoes with a total area of 0.03 m², the same person gives 600 ÷ 0.03 = 20,000 Pa. The heel's pressure is 300 times bigger, which is why heels dent soft floors.

Example 3: force from air pressure

What force does the atmosphere put on a 1.5 m² table top? Choose Force, type P = 101325 (or pick atm as the pressure unit and type 1) and A = 1.5.

About 152 kN, the weight of a few cars. The table doesn't collapse because air pushes up on its underside with almost the same force.

Example 4: pressure on a diver

A diver is 12 m below the surface of the sea. Choose Fluids and Pressure, pick Seawater (1,025 kg/m³) from Common values and type h = 12.

That is the gauge pressure from the water. With the atmosphere on top reads 221,988 Pa, the absolute pressure on the diver: about 2.19 atm, more than twice the pressure at the surface.

Example 5: depth from a pressure reading

A sensor at the bottom of a freshwater tank reads a gauge pressure of 50 kPa. How deep is the water? Choose Fluids and Depth, set the pressure unit to kPa, type 50 and pick Water.

Example 6: a mercury barometer

How tall a column of mercury does the atmosphere hold up? Choose Depth again, type P = 101325 and pick Mercury (13,534 kg/m³). The calculator gives h = 101325 ÷ (13534 × 9.81) = 0.763 m, or 763 mm. The textbook figure is 760 mm because the millimetre of mercury is defined with a density of 13,595.1 kg/m³ and g = 9.80665 m/s²: 101325 ÷ (13595.1 × 9.80665) = 0.760 m. The calculator's mercury, at room temperature, is a little less dense.

Example 7: which liquid is it?

At 3 m down in a tank of an unknown liquid, the gauge pressure is 22,000 Pa. Choose Fluids and Density, then type P = 22000 and h = 3: ρ = 22000 ÷ (9.81 × 3) = 747.5 kg/m³. That is very close to petrol's 750 kg/m³, and well below water's 1,000.

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What the calculator doesn't do

• Gas pressure from temperature or volume. For PV = nRT or Boyle's law, use the Gas Laws Calculator, explained in the Boyle's law calculator guide.
• A force at an angle. P = F/A uses the force at right angles to the surface. If the force is at angle θ to the surface, use F sin θ as the force.
• Area from a shape. Work out the area first, for example π r² for a circular piston, then type it in.
• Square inches. Area units go up to ft², not in². For psi problems with areas in square inches, P = F ÷ A in lbf and in² gives psi directly, by hand.

Common mistakes

• Mixing cm² and m². 1 m² is 10,000 cm², not 100. Pick the right area unit, or convert before you divide.
• Using mass instead of weight. F is a force in newtons. A 60 kg person weighs 60 × 9.81 = 588.6 N.
• Forgetting the atmosphere. ρgh is gauge pressure. Add 101,325 Pa if the question asks for the total pressure.
• Measuring depth from the bottom. h is the depth below the surface of the fluid.
• Using the width of the tank. Pressure at a depth doesn't depend on the container's shape or size.

Read next

Chemistry · Rates and equilibriumPartial pressure: formula, mole fraction and KpPartial pressure explained: Dalton's law, mole fraction × total pressure, how to write and calculate Kp for A-level Chemistry, with worked examples.