Pressure Calculator

Definition of pressure and the pascal

Pressure p is defined as the ratio of the force F acting perpendicular to a surface to the area A of that surface: p = F/A. It is a scalar — unlike force it has no direction, although the force that produces it does. If the force is not distributed evenly, the quotient gives the average pressure; the pressure at a point is defined by the limit p = dF/dA as the area shrinks to zero.

The SI unit of pressure is the pascal, 1 Pa = 1 N/m² = 1 kg·m⁻¹·s⁻². The name was adopted by the 14th General Conference on Weights and Measures in 1971 in honor of Blaise Pascal. The pascal is small on an everyday scale: standard atmospheric pressure at sea level is about 101,325 Pa, which is why practice uses hectopascals (meteorology), kilopascals (engineering), and megapascals (strength of materials). In the United States the pound-force per square inch (psi) remains common; 1 psi = 6,894.757 Pa.

Historical background: Torricelli, Pascal, and the unit’s name

The quantitative notion of pressure grew out of studies of the atmosphere. In 1643 Evangelista Torricelli built the mercury barometer and showed that air exerts a pressure balanced by a mercury column about 76 cm high. In 1648 Blaise Pascal organized a measurement on the Puy de Dôme (carried out by his brother-in-law Florin Périer): the mercury column shortened with altitude, confirming that pressure comes from the weight of the air.

Pascal’s posthumously published Traité de l’équilibre des liqueurs (1663) states the principle now called Pascal’s law: pressure applied to a confined fluid is transmitted equally in all directions. That principle is behind the hydraulic press and the bottle jack that appear among the examples on this page. Pascal’s name entered the SI only three centuries later.

Assumptions of the model p = F/A: normal force and uniform distribution

The formula p = F/A correctly describes a force acting perpendicular to a surface and spread evenly over it. When the force acts at an angle θ to the normal, only its normal component F·cos θ produces pressure, while the tangential component F·sin θ produces shear stress, not pressure. This calculator assumes θ = 0, so the value in the Force field should already be the perpendicular component.

The second assumption concerns uniformity. A foot on sand, a tire on asphalt, or a vise jaw on wood does not press every square millimeter of the contact equally. The calculator’s result is then the average pressure: local values can be noticeably higher where the load concentrates. For school problems and engineering estimates the average pressure is usually the right quantity; for a failure analysis of the material the full contact-pressure distribution is needed.

Derivation and the link to hydrostatics

Step 1. A force F acts perpendicular to a flat surface of area A. By definition p = F/A; in units, [N]/[m²] = [Pa]. Step 2. If the force is the weight of a body of mass m, then F = m·g, so p = m·g/A. Seventy kilograms at g = 9.81 m/s² give about 687 N — hence 700 N as a rounded example of an adult’s weight.

Step 3. The hydrostatic formula is a special case of the same definition. Consider a column of liquid with density ρ, height h, and base area A. Its mass is m = ρ·A·h, its weight F = ρ·A·h·g, and the pressure on the base F/A = ρ·g·h. The area cancels, which is why pressure at depth h does not depend on the shape of the container — the so-called hydrostatic paradox. Step 4. Inverting this relation, the calculator reports the height of a water column h = p/(ρ·g) that would exert the computed pressure; for 35 kPa and water at 1,000 kg/m³ that is about 3.57 m, or 11.7 ft.

Pressure versus stress: the Cauchy tensor and Hertzian contact

In continuum mechanics the internal forces are described by the Cauchy stress tensor σ (Augustin-Louis Cauchy, 1822). Mechanical pressure is its isotropic part: p = −(1/3)·tr σ = −(σ₁₁ + σ₂₂ + σ₃₃)/3, with the convention that tension is positive and compression negative. In a fluid at rest the shear stresses vanish and the tensor reduces to −p·I, so the pressure is the same in every direction — Pascal’s law in tensor language.

The contact of two solids over a small patch is not uniform. Heinrich Hertz, in Über die Berührung fester elastischer Körper (1882), derived that for a sphere pressed against a plane the contact radius grows as F^(1/3) and the maximum pressure at the center of the patch is 1.5 times the mean pressure F/(π·a²). This theory is applied to bearings, wheels, and gear teeth. For this page it means one thing: p = F/A gives the average, and the true maximum under a nail or a thumbtack is higher still.

Units, conversions, and gauge pressure

Conversion factors used by the calculator: 1 bar = 100,000 Pa (exactly), 1 atm = 101,325 Pa (exactly, by definition of the standard atmosphere), 1 psi = 6,894.757 Pa (pound-force per square inch), 1 mmHg = 133.322 Pa, 1 Torr = 101,325/760 Pa ≈ 133.322 Pa, 1 hPa = 100 Pa. These values appear in the BIPM SI Brochure and NIST Special Publication 811; NIST recommends the pascal and lists other units only as conversions.

Engineering distinguishes absolute pressure (measured from vacuum) from gauge pressure (measured from atmospheric pressure). The gauge on a compressor or a tire reads gauge pressure: “32 psi in a tire” means roughly 47 psi (about 320 kPa) absolute. The quotient p = F/A in this calculator is simply the pressure a force exerts on a surface; it does not add atmospheric pressure, which acts on both sides of most objects and cancels in the balance.

Orders of magnitude for orientation on the logarithmic scale: 1 Pa is roughly the pressure of a dollar bill lying flat on a table; 10–100 kPa is the pressure under a human foot; 101.325 kPa is one atmosphere; 200–250 kPa (29–36 psi) is a passenger-car tire gauge pressure; about 110 MPa (16,000 psi) acts at the bottom of the Mariana Trench (depth about 11 km); 1 GPa is the order of pressure in diamond-synthesis presses.

Sources

The definition, unit, and derivation come from general-physics textbooks and metrology documents: D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics (chapter on fluids), Wiley; H. D. Young, R. A. Freedman, University Physics, chapter “Fluid Mechanics”; R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. II, chapters 31 (tensors) and 38–39 (elasticity); L. D. Landau, E. M. Lifshitz, Theory of Elasticity; K. L. Johnson, Contact Mechanics, Cambridge University Press, 1985; H. Hertz, Über die Berührung fester elastischer Körper, Journal für die reine und angewandte Mathematik, vol. 92, 1882; B. Pascal, Traité de l’équilibre des liqueurs, 1663; BIPM, The International System of Units (SI), 9th edition; NIST SP 330 and SP 811. This page is for educational purposes.