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Pressure (force per unit area)

Pressure is the amount of force applied perpendicular to a surface per unit area. It is fundamental in physics, engineering, meteorology and physiology, and is measured in pascals and other units.

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Overview

Pressure describes how strongly a force pushes against a surface relative to the size of that surface. In its simplest form it is expressed by the formula P = F / A, where P is pressure, F is the normal (perpendicular) component of the force and A is the area over which that force is distributed. This basic definition helps distinguish pressure from force: a large force spread over a large area can produce less pressure than a smaller force concentrated on a tiny area.

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Key characteristics and formulas

Common formulas and relations used for pressure include:

  • P = F / A (force per unit area)
  • Hydrostatic pressure in a fluid column: p = ρ g h, where ρ is fluid density, g is gravitational acceleration and h is depth.
  • Ideal‑gas relation linking pressure, volume and temperature: pV = nRT (for ideal gases).

Pressure acts perpendicular to a surface and is a scalar quantity in continuum mechanics (it does not have direction but it is associated with a normal force acting on a surface). When liquids and gases are considered, pressure at a point acts equally in all directions in the absence of shear stresses.

Units and measurement

The International System of Units (SI) uses the pascal (Pa) as the base unit: 1 pascal equals 1 newton per square meter (1 N/m2). Practical engineering and everyday contexts often use multiples and alternatives such as kilopascals (kPa), megapascals (MPa), bar, atmospheres (atm) and pounds per square inch (psi). Instruments that measure pressure include manometers, gauges and the barometer — historically associated with Torricelli and later used by Blaise Pascal — which measure atmospheric pressure and its variations.

Hydraulics, fluids and Pascal’s principle

In fluids, pressure transmits through the medium. Pascal’s principle states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and to the walls of its container. This principle underpins hydraulic machinery such as lifts and brakes, where a small force applied on a small-area piston produces a larger force on a larger-area piston by converting pressure into force across areas. The behavior of fluids also links pressure to fluid weight and density: a denser fluid or greater depth produces larger hydrostatic pressure.

Applications and examples

Pressure is central to many fields and everyday phenomena. Examples include:

  • Meteorology: atmospheric pressure differences drive winds and weather systems; high- and low-pressure areas influence cloud formation and storms (atmospheric pressure observations).
  • Engineering: structural design, material strength and sealing depend on pressure loads and safety margins; hydraulic systems use pressure to transmit force.
  • Medicine: blood pressure is a vital health indicator measured as systolic/diastolic values using sphygmomanometers and clinical devices.
  • Diving and aviation: pressure changes with depth and altitude affect human physiology and equipment performance; pressure differentials are critical for cabin pressurization and decompression considerations.

Distinctions and notable facts

Important distinctions include absolute pressure versus gauge pressure. Absolute pressure is measured relative to a perfect vacuum (vacuum), while gauge pressure is measured relative to ambient atmospheric pressure. A related concept is thrust, often used to describe compressive force acting normally on a surface. Density and weight influence pressure when an object's mass produces force distributed over an area: heavier materials or deeper fluid columns create larger pressures at the same area and geometry (force, density).

Pressure combines simple mathematical form with wide-ranging practical consequences. From keeping buildings standing to forecasting weather and designing medical devices, understanding how forces distribute over areas is essential in science and technology. For introductions, measurement standards and historical context see introductory references and instrument descriptions such as the classic barometer and modern pressure gauges (barometer, definition, pascal).

Here the force F is perpendicular to the surface A
Here the force F is perpendicular to the surface A

History

In antiquity, Archimedes, Ctesibios, Philon of Byzantium, Heron of Alexandria and Sextus Iulius Frontinus were already aware of the effect of the pressure of water and air. In the Middle Ages, Alhazen should be mentioned, who had a correct idea of air pressure before in the Renaissance the Dutch merchant Simon Stevin (1548-1620) formulated the first principles of hydrostatics and the hydrostatic paradox, see picture.

Fundamental research began in the 17th century at the court of Grand Duke Cosimo II de' Medici. There, the well master was astonished to discover that he could not lift water higher than 32 feet (10.26 m) using a suction pump. An airless space formed above the column of water - as in the pipe in the picture in the area A-C - which prevented it from rising any higher. This phenomenon was reported to Cosimo II's teacher and court mathematician, Galileo Galilei, who subsequently treated it in his Discorsi (pp. 16-17). Vincenzo Viviani, an associate of Galileo, was the first to conclude in 1643 that it is air pressure that pushes the water up the suction tube (pictured at B). Evangelista Torricelli, Galileo's assistant and successor, made experiments with a tube filled with mercury as in the picture and explained from the difference in density of water and mercury why the former rises 13½ times higher than the latter at 760 mm. Thus Torricelli invented the mercury barometer.

The news of the "Italian experiment" came to Blaise Pascal in 1644 via Marin Mersenne and the physicist Pierre Petit. The latter repeated Torricelli's experiments and concluded that the pressure in a liquid or gas is proportional to its depth. Accordingly, if the column of mercury is carried by atmospheric pressure, its height on a mountain must be less than in the valley. Petit and Pascal's brother-in-law Florin Périer made the corresponding measurements at Clermont-Ferrand and on the summit of the 1465 m high Puy de Dôme on 19 September 1648 and obtained the expected results. Already in October Pascal published his results as a report of the great experiment on the equilibrium of liquids (Pascal: Récit de la grande expérience de l'équilibre des liqueurs). In the treatise on the equilibrium of liquids and on the weight of the mass of air of 1653, Pascal formulated among other things

  • Pascal's principle, according to which pressure spreads out in all directions in liquids at rest,
  • Pascal's law for hydrostatic pressure, which increases linearly with depth, see below, and
  • the operating principle of a new machine for multiplying forces (Pascal: machine nouvelle pour multiplier les forces), i.e. the hydraulic press.

Otto von Guericke demonstrated his famous experiment with the Magdeburg hemispheres in front of the Reichstag in Regensburg in 1654, see picture.

New findings came from, among others

  • Robert Boyle and Edme Mariotte in 1662 by the act of Boyle-Mariotte,
  • Daniel Bernoulli in 1738 by tracing the pressure of gases back to the collisions of the gas molecules (kinetic theory of gases) and by distinguishing between hydrostatic and hydrodynamic pressure (Bernoulli pressure equation),
  • Leonhard Euler with his definition of pressure within a fluid in its still valid form, i.e. that it acts within the volume in all directions but always perpendicular to walls, and
  • John Dalton 1802 by discovery of the partial pressures of gases in gas mixtures (Dalton's law)
Hydrostatic paradox according to Stevin: The little water in the area ABCD of the vessel presses against the wall CD just as strongly as the much water in CDEF.
Hydrostatic paradox according to Stevin: The little water in the area ABCD of the vessel presses against the wall CD just as strongly as the much water in CDEF.
Stitch of Guericke's hemisphere attempt
Stitch of Guericke's hemisphere attempt

Definition

Pressure is the result of a force acting on a surface. The magnitude of the pressure on the reference surface A results exclusively from the force component F_{n}perpendicular to the surface. Mathematically for a plane surface A:

{\displaystyle p={\frac {F_{n}}{A}}}

For curved surfaces or location-dependent pressure, a sufficiently small area element dA must be considered:

{\displaystyle p=\lim _{\mathrm {d} A\to 0}{\frac {\mathrm {d} F_{n}}{\mathrm {d} A}}}

with:

p

- squeeze

{\displaystyle \mathrm {d} F_{n}}

- Normal force and

\mathrm {d} A

- Surface on which the force acts.

Vectorially, the pressure is the constant of proportionality between the vector surface element \mathrm{d}\vec Aand the normal force {\displaystyle \mathrm {d} {\vec {F}}_{n},}acting on this element:

{\displaystyle \mathrm {d} {\vec {F}}_{n}=-p\,\mathrm {d} {\vec {A}}=-p\,{\hat {n}}\,\mathrm {d} A}.

The normal unit vector {\hat {n}}on the surface is parallel to the force and here points outward away from the body. The minus sign causes a positive pressure when the force is directed toward the body. A compressive force acts antiparallel to this outward normal vector, that is, toward the body (a force acting outward in the direction of the normal vector is a tensile force).

Occasionally it is said that pressure acts in a certain direction. Physically, it would be more correct to speak of the pressure force, which can press in one direction. In physics, however, pressure as a scalar quantity is directionless or "acting in all directions".

For incompressible and for compressible fluids, different components contribute to the total pressure. For free-flowing fluids, incompressibility can be assumed to a good approximation at velocities far below the wave propagation velocity, especially in liquids. Quiescent gases, on the other hand, are compressible.

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