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How to Calculate Pressure
Pressure is force per unit area. The formula is:
\[P = \frac{F}{A}\]Where P is pressure, F is force in newtons, and A is the area over which that force is distributed in square meters. The SI unit of pressure is the pascal (Pa), equal to one newton per square meter. A person weighing 700 N standing on a floor exerts a pressure of 700 / 0.06 = approximately 11,667 Pa on the ground, assuming a combined shoe contact area of about 0.06 square meters (roughly the area of two shoe soles).
Pressure units
Pressure appears in several different units depending on the field. The table below shows the most common ones and their conversions.
| Unit | Symbol | Equivalent to 1 atm |
|---|---|---|
| Pascal | Pa | 101,325 Pa |
| Kilopascal | kPa | 101.325 kPa |
| Pounds per square inch | PSI | 14.696 PSI |
| Bar | bar | 1.01325 bar |
| Atmosphere | atm | 1 atm |
| Millimeters of mercury | mmHg | 760 mmHg |
| Torr | Torr | 760 Torr |
One atmosphere is the approximate pressure of Earth’s atmosphere at sea level. PSI is common in the United States for tire pressure and industrial applications. Kilopascals and bar are standard in most other countries and in engineering. Millimeters of mercury and torr appear in medical and laboratory settings (blood pressure readings use mmHg). All of these units measure the same physical quantity; the numbers just differ by conversion factors.
Worked example: a person standing
A person has a mass of 75 kg. The gravitational force (weight) is:
\[F = m \times g = 75 \times 9.81 = 735.75 \text{ N}\]Standing in flat shoes, the combined contact area of both feet is about 0.05 square meters. The pressure on the floor is:
\[P = \frac{735.75}{0.05} = \textbf{14,715 Pa}\]That is approximately 14.7 kPa, or about 2.13 PSI. Now consider the same person wearing stiletto heels, where the heel tip has an area of about 0.0001 square meters (1 square centimeter). If half the person’s weight rests on one heel:
\[P = \frac{367.9}{0.0001} = \textbf{3,679,000 Pa}\]That is roughly 3.68 MPa, or about 534 PSI. The force has not changed; only the area changed. Concentrating the same force over a smaller area produces dramatically higher pressure, which is why stiletto heels can damage soft flooring. The force calculator can help determine the force component if you know mass and acceleration.
Worked example: a hydraulic press
Hydraulic systems use pressure to multiply force. A small piston with an area of 0.002 square meters receives a force of 500 N, creating a pressure of:
\[P = \frac{500}{0.002} = 250{,}000 \text{ Pa} = 250 \text{ kPa}\]That pressure transmits equally through the hydraulic fluid to a large piston with an area of 0.04 square meters. The force on the large piston is:
\[F = P \times A = 250{,}000 \times 0.04 = \textbf{10,000 N}\]A 500 N input produced a 10,000 N output, a multiplication factor of 20. The factor equals the ratio of the large piston area to the small piston area (0.04 / 0.002 = 20). Car brakes, hydraulic jacks, and heavy industrial presses all rely on this principle.
Worked example: tire pressure
A car tire is typically inflated to about 32 PSI, which is approximately 220.6 kPa or 2.21 bar. If the tire contact patch (the area of rubber touching the road) is about 0.045 square meters, and the tire pressure supports one-quarter of a 1,600 kg car:
\[F = \frac{1{,}600 \times 9.81}{4} = 3{,}924 \text{ N per tire}\] \[P = \frac{3{,}924}{0.045} = 87{,}200 \text{ Pa} = 87.2 \text{ kPa}\]This is the contact pressure between the tire and the road. It differs from the inflation pressure (220.6 kPa) because the tire structure distributes load across the contact patch in a more complex way than a simple balloon would. Still, the basic formula applies: more weight on the tire or a smaller contact patch increases the contact pressure.
Gauge pressure vs. absolute pressure
Pressure gauges on tires, compressors, and boilers typically read zero when exposed to the atmosphere. A tire gauge showing 32 PSI means 32 PSI above atmospheric pressure. This is gauge pressure.
Absolute pressure is gauge pressure plus atmospheric pressure:
\[P_{\text{absolute}} = P_{\text{gauge}} + P_{\text{atmospheric}}\]At sea level, atmospheric pressure is approximately 14.7 PSI (101.325 kPa). A tire at 32 PSI gauge has an absolute pressure of 32 + 14.7 = approximately 46.7 PSI (322 kPa).
The distinction matters in scientific calculations, gas law problems, and any application where the reference point affects the result. The ideal gas law (PV = nRT) uses absolute pressure. A “vacuum” in everyday language means low absolute pressure, not negative gauge pressure, though gauge readings can display negative values when the measured pressure is below atmospheric.
Hydrostatic pressure in fluids
For stationary fluids, pressure increases with depth. The formula is:
\[P = \rho g h\]Where P is the pressure due to the fluid column (not including atmospheric pressure above the surface), rho is the fluid density in kg per cubic meter, g is gravitational acceleration (9.81 m/s squared), and h is the depth in meters.
Water has a density of approximately 1,000 kg per cubic meter. At a depth of 10 meters:
\[P = 1{,}000 \times 9.81 \times 10 = \textbf{98,100 Pa}\]That is approximately 98.1 kPa, or roughly 0.97 atm. Every 10 meters of water depth adds approximately one atmosphere of pressure. A diver at 30 meters experiences about 3 atm of hydrostatic pressure plus 1 atm of atmospheric pressure at the surface, for a total absolute pressure of about 4 atm.
This same principle explains why water towers work. Elevating the water creates hydrostatic pressure at ground level without any pump running. A water tower 40 meters tall produces a pressure of approximately 392,400 Pa (about 56.9 PSI) at its base, which is sufficient to push water through residential plumbing.
Atmospheric pressure
The air above us has weight, and that weight pressing down creates atmospheric pressure. At sea level, the standard atmosphere is 101,325 Pa (101.325 kPa, 14.696 PSI, or 760 mmHg). This value decreases with altitude because there is less air above. At the summit of Mount Everest (8,849 meters), atmospheric pressure falls to roughly 33.7 kPa, about one-third of sea-level pressure.
Common mistakes
The most frequent error is using inconsistent units. If force is in newtons and area is in square centimeters, the result is not in pascals. Convert area to square meters first, or convert the result afterward. One square centimeter is 0.0001 square meters.
Another common mistake is confusing gauge pressure with absolute pressure when using gas law equations. If a problem provides gauge pressure, add atmospheric pressure before plugging values into PV = nRT.
Finally, in hydrostatic problems, remember that P = rho times g times h gives only the pressure from the fluid column. The total pressure at depth also includes whatever pressure exists at the surface, typically atmospheric pressure for an open container.