Pressure — P = F/A

The pressure formula P = F/A explained: force spread over area, pascals, why sharp things cut, and a worked example.

P = \frac{F}{A}

Pressure equals force divided by the area it acts on.

What it means

Pressure is force with its footprint taken into account. The same push can be a gentle press or a piercing stab depending on the area it acts through — pressure captures that difference in one number. One pascal is one newton per square meter, a tiny unit, so everyday pressures are quoted in kilopascals: atmospheric pressure is about 101 kPa, a car tire around 220 kPa. In fluids, pressure also grows with depth ( $P = \rho gh$ ), because each layer must support the weight of everything above it.

Variables

Symbol	Name	Unit	Meaning
$P$	Pressure	Pa	Force per unit area, perpendicular to the surface
$F$	Force	N	Magnitude of the force acting perpendicular to the surface
$A$	Area	m²	Area over which the force is spread

Assumptions

Force acts perpendicular to the surface
Pressure is uniform over the area considered
Static or quasi-static conditions (for dynamic pressure see Bernoulli)

Derivation

Pressure is defined as the normal force per unit area: $P = dF_\perp/dA$ . For a uniform force over a flat area this becomes the simple ratio $F/A$ . In a fluid at rest, pressure acts equally in all directions (Pascal's principle).

Worked example

Problem

A 600 N force presses straight down on a 0.02 m² piston. What pressure does it exert?

Given:

F = 600 N
A = 0.02 m²

Write the definition: $P = F/A$
Substitute: $P = 600 / 0.02$
Divide: $P = 30{,}000\ \mathrm{Pa}$

30,000 Pa = 30 kPa

Six hundred newtons over two-hundredths of a square meter gives 30 kilopascals — about a third of atmospheric pressure. Spread the same force over a larger area and the pressure falls; concentrate it on a pinhead and it skyrockets, which is why sharp knives cut and snowshoes don't sink.