Formula of Pressure: P = F/A, SI Unit, ρgh and Examples

The formula of pressure is $P = \frac{F}{A}$ : the force acting at right angles to a surface divided by the area that carries it. One newton spread over one square metre is one pascal (Pa), the SI unit.

The same force can give tiny or huge pressures. A boot spreads your weight and you stand on snow; a skate blade concentrates it and you sink a little. Inside a liquid, the weight of the liquid above produces pressure too, and that version, $P = \rho g h$ , links this page to density and the rest of the fluids formula sheet.

Quick facts

Formula
$P = \frac{F}{A}$
SI unit
pascal, 1 Pa = 1 N/m²
Dimensional formula
[M¹ L⁻¹ T⁻²]
Type
Scalar: it pushes equally in all directions at a point in a fluid
In a liquid
$P = P_0 + \rho g h$
Other units
1 atm = 1.013 × 10⁵ Pa, 1 bar = 10⁵ Pa, 1 kPa = 1000 Pa

Pressure formula in physics, term by term

Quantities in P = F / A
Symbol	What it is	SI unit
$P$	Pressure on the surface	Pa
$F$	Normal (perpendicular) component of the force	N
$A$	Contact area	m²

Only the perpendicular part of the force counts. If a force pushes at an angle θ to the surface, use $F\cos\theta$ where θ is measured from the normal. A sideways drag along the surface makes shear stress, not pressure.

Pressure is a scalar even though force is a vector. At any point inside a still fluid it acts equally in every direction, which is why a dam wall feels the push sideways and a submarine hull is squeezed from all sides.

What is the formula of pressure in a liquid

Picture a column of liquid of height h and cross section A. Its mass is $\rho A h$ and its weight is $\rho A h g$ . Divide by the area:

$P = \frac{\rho A h g}{A} = \rho g h$

This is the gauge pressure, the extra pressure due to the liquid alone. With the air pressure $P_0$ above the surface, the absolute pressure at depth h is

$P = P_0 + \rho g h$

The shape of the container does not matter, only the vertical depth. A wide lake and a narrow pipe filled to the same height give the same pressure at the bottom. The same depth formula explains the upward push on submerged objects in the buoyant force formula.

Solved examples

Example 1: standing on the floor

A student weighing 600 N stands on both feet. The total contact area of the soles is 0.040 m². What pressure does the student exert?

$P = \frac{600}{0.040} = 1.5 \times 10^4\ \text{Pa}$

Standing on one foot halves the area and doubles the pressure to 3.0 × 10⁴ Pa.

Example 2: a drawing pin, with a unit conversion

A thumb pushes a drawing pin with 20 N. The point has an area of 0.10 mm². Find the pressure at the point.

Convert: 1 mm² = 10⁻⁶ m², so A = 0.10 × 10⁻⁶ = 1.0 × 10⁻⁷ m².

$P = \frac{20}{1.0 \times 10^{-7}} = 2.0 \times 10^8\ \text{Pa}$

That huge pressure is why the pin enters the board while the thumb, pressing on the wide head, is unharmed.

Example 3: a diver in the sea

Find the gauge and absolute pressure on a diver 12 m below the surface of sea water (density 1025 kg/m³). Take atmospheric pressure as 1.013 × 10⁵ Pa.

Gauge: $P = \rho g h = 1025 \times 9.8 \times 12 = 1.21 \times 10^5\ \text{Pa}$

Absolute: $P = 1.013\times10^5 + 1.205\times10^5 = 2.22 \times 10^5\ \text{Pa}$ , a little over twice the pressure at the surface.

Area conversions square the length factor: 1 cm² = 10⁻⁴ m² and 1 mm² = 10⁻⁶ m².

Units and dimensions

Force has dimensions [M L T⁻²] and area [L²]. Dividing gives [M¹ L⁻¹ T⁻²], worked out in full on the dimensional formula of pressure page. Stress, Young's modulus and energy per unit volume share these dimensions, which is a useful cross check.

Pressure units you will meet
Unit	Value in pascals	Where it appears
kilopascal (kPa)	1000 Pa	tyre gauges, weather data
bar	1.0 × 10⁵ Pa	engineering, cylinders
atmosphere (atm)	1.013 × 10⁵ Pa	chemistry, gas laws
mm of mercury (mmHg)	133 Pa	blood pressure, barometers

Mistakes to avoid

Using mass for F. Convert kilograms to a weight in newtons with W = mg.
Leaving area in cm². Divide by 10⁴ to get m² before substituting.
Mixing gauge and absolute pressure. Read whether the question wants the total, which includes air pressure.
Taking a slant length for h. In ρgh, h is the vertical depth below the free surface.
Adding pressures from different areas as forces. Convert each to force with F = PA first, then add.

Frequently asked questions

What is the formula of pressure in physics?

Pressure is force per unit area, P = F / A, with F perpendicular to the surface. In a fluid at depth h it is P = P₀ + ρgh.

Which unit is pressure measured in?

The pascal, Pa, equal to one newton per square metre. Atmospheric pressure is about 1.013 × 10⁵ Pa.

Is pressure a vector or scalar?

Scalar. At a point in a fluid it has no direction of its own; only the force it exerts on a chosen surface has one, along the normal.

How does area affect pressure?

For a fixed force, pressure is inversely proportional to area. Halve the area and the pressure doubles, which is why knives are sharpened and tractors have wide tyres.

What is the difference between pressure and stress?

They share a unit. Pressure describes a fluid or a contact pushing on a surface, always along the normal; stress describes the forces inside a solid that resist deformation, and it can act sideways as shear.