Pressure Calculator

Results

How to Use the Pressure Calculator

1. 1. Choose the kind of pressure. Force on an area solves P = F/A. Depth in a fluid solves P = ρgh for the pressure, the depth or the fluid's density.
2. 2. Pick the unknown and fill in the rest. The box for the unknown becomes the answer. Each box has a unit menu (Pa, kPa, bar, atm, psi, mmHg; cm², mm², in²; and so on). Scientific notation such as 1.2e5 works. The example and fluid menus load realistic values.
3. 3. For fluids, choose gauge or absolute. Gauge pressure is what a tyre gauge reads, the amount above the atmosphere. Absolute pressure counts from a vacuum, so it adds the pressure on the surface, P₀ (standard atmosphere is 101,325 Pa).
4. 4. Read the results. You get gauge and absolute pressure, the depth that adds one atmosphere in this fluid, and the working in SI units. To convert a pressure into other units, use the Pressure Unit Converter.

The Pressure Formulas

P = F / A

P_gauge = ρ g h

P_abs = P₀ + ρ g h

Pressure is force per unit area, measured in pascals: 1 Pa = 1 N/m². The force must act perpendicular to the surface, and it is assumed to spread evenly. The same force on a smaller area gives a larger pressure, which is why a knife cuts, a thumbtack pierces and a snowshoe stays on top of the snow.

In a fluid at rest, pressure rises with depth because each layer holds up the fluid above it: P = ρgh, where ρ is the density, g the gravitational acceleration and h the depth below the surface. The pressure depends only on depth, not on the width or shape of the container. Add the pressure on the surface, usually the atmosphere, to get the absolute pressure. See OpenStax University Physics, section 14.1.

Worked Examples

1. A person on both feet. A 70 kg person weighs 686.5 N. Standing on 400 cm² (0.04 m²) of shoe sole, the pressure is 17,160 Pa, about 0.17 atm.
2. The same person on one stiletto heel. On 1 cm² the pressure is 6.865 × 10⁶ Pa, about 68 atmospheres: 400 times higher, from the same force on one four-hundredth of the area.
3. A thumbtack. Pushed with 10 N onto a tip of 0.1 mm², it exerts 1.000 × 10⁸ Pa, or 100 MPa.
4. Ten metres under fresh water. With ρ = 998.2 kg/m³ and g = 9.80665 m/s², P_gauge = 97,890 Pa and P_abs = 101,325 + 97,890 = 199,200 Pa, which is 1.966 atm. In seawater (1025 kg/m³) the gauge pressure at the same depth is 100,500 Pa.
5. A rule of thumb. Each additional atmosphere takes 10.35 m of fresh water, or 10.08 m of seawater. A diver at 30 m is at 3.898 atm absolute, almost four times the pressure at the surface. On the Moon, where g is 1.62 m/s², it takes 62.7 m of water to add an atmosphere.

Why a Mercury Barometer Reads 760 mm

Set the fluid to mercury and solve for depth at one atmosphere of gauge pressure. Mercury at 20 °C (13,546 kg/m³) gives 0.7628 m, about 763 mm. The familiar 760 mmHg is slightly lower because the millimetre of mercury is defined with mercury at 0 °C, 13,595.1 kg/m³, which is denser. The same calculation with air (1.225 kg/m³) gives 8,435 m, the height of an atmosphere of uniform density. The real atmosphere thins with height, so it extends much higher.

Assumptions and Limits

• Incompressible, uniform fluid. Water is about 5% denser at the bottom of the deepest ocean trenches, and gases compress a great deal, so ρgh is a good model for liquids over modest depths and a poor one for a tall column of gas.
• Constant g. Fine over any depth in a pool or ocean; the calculator lets you switch to another planet's gravity.
• Stated densities. Preset densities are representative values at about 20 °C. Real densities shift with temperature, salinity and purity.
• Force perpendicular to the surface. For an angled force, use only the component at right angles to the surface.

Frequently Asked Questions

What is the formula for pressure?

P = F / A: pressure equals the force divided by the area over which it acts, in pascals when the force is in newtons and the area in square metres. For pressure in a fluid at depth h, use P = ρgh, then add the surface pressure for the absolute value.

What is the difference between gauge and absolute pressure?

Gauge pressure is measured relative to the surrounding atmosphere; absolute pressure is measured relative to a perfect vacuum. Absolute = gauge + atmospheric, so a gauge reading of 0 means atmospheric pressure, about 101,325 Pa at sea level.

How do you calculate pressure at a depth in water?

Multiply the density of water (about 998 kg/m³ for fresh water, 1025 for seawater) by g (9.80665 m/s²) and the depth in metres. At 10 m that gives about 98 kPa of gauge pressure, so the absolute pressure is nearly 2 atmospheres.

Does pressure depend on the shape or size of the container?

No. In a fluid at rest, pressure depends only on depth, density and g. A narrow tube and a wide lake give the same pressure at the same depth, which is why the water in connected vessels settles to one level.

Why does a smaller area give a higher pressure?

Because pressure is force divided by area, shrinking the area while the force stays the same increases the pressure in inverse proportion. Halving the area doubles the pressure.

What units is pressure measured in?

The SI unit is the pascal (Pa), equal to 1 N/m². Common units include the kilopascal, the bar (100,000 Pa), the standard atmosphere (101,325 Pa), pounds per square inch and millimetres of mercury (133.322 Pa). The Pressure Unit Converter converts between them.