Dimensional Formula of Pressure: [ML⁻¹T⁻²] Derived from F/A

The dimensional formula of pressure is [M¹ L⁻¹ T⁻²], often printed as [ML⁻¹T⁻²]. Pressure grows with mass to the first power, shrinks with length to the first power and falls with the square of time.

The result comes straight from the definition: pressure is a normal force spread over an area. Force carries [MLT⁻²] and area carries [L²], so dividing removes one more length. The working, the unit, the look alike quantities and a quick check on a liquid column all follow below. For numbers and the full set of pressure equations, open the pressure formula page.

Quick facts

Dimensional formula
[M¹ L⁻¹ T⁻²], short form [ML⁻¹T⁻²]
Defining formula
$P = \frac{F}{A}$
SI unit
pascal (Pa) = N/m² = kg m⁻¹ s⁻²
CGS unit
dyne/cm², where 1 Pa = 10 dyne/cm²
Scalar or vector
Scalar
Same dimensions as
Stress, elastic moduli, energy density

Derivation of the dimensional formula of pressure

Start from what pressure means, then swap every symbol for its base dimensions.

Definition: $P = \frac{F}{A}$ , normal force per unit area.
Force from Newton\'s second law: $[F] = [M][L T^{-2}] = [M L T^{-2}]$ .
Area is length times length: $[A] = [L^{2}]$ .
Divide: $[P] = [M L T^{-2}] \div [L^{2}]$ , which leaves $[M L^{-1} T^{-2}]$ .

So the dimensional formula of pressure is [M¹ L⁻¹ T⁻²]. Mass and time keep the same powers they have in force. Only the length power changes, from +1 to −1, because the area in the denominator brings in L⁻². The dimensional formula of force page shows how [MLT⁻²] itself is built.

From dimensions to the pascal

Read each letter of [M¹ L⁻¹ T⁻²] as its SI base unit: kilogram for M, metre for L, second for T. That gives kg m⁻¹ s⁻². The same thing is a newton per square metre, and the SI gives it the special name pascal (Pa).

$1\ \text{Pa} = 1\ \text{N/m}^2 = 1\ \text{kg m}^{-1}\,\text{s}^{-2}$

Other pressure units are just multiples of the pascal, so they share the dimensions:

Common pressure units, all with dimensions [ML⁻¹T⁻²]
Unit	Value in pascals
kilopascal (kPa)	10³ Pa
bar	10⁵ Pa
standard atmosphere (atm)	1.013 × 10⁵ Pa
dyne per square centimetre (CGS)	0.1 Pa

A unit such as mm of mercury hides the density and g of the liquid inside it. It still measures pressure, so its dimensions are still [ML⁻¹T⁻²].

Other quantities that share pressure's dimensions

Several quantities in elasticity, fluids and fields reduce to the same bracket as pressure. Spotting them saves time in match the column questions.

Quantities with the dimensions [M¹ L⁻¹ T⁻²]
Quantity	Defining relation	Reason it matches
Stress	$\sigma = \frac{F}{A}$	Internal force per area, same build as pressure
Young's modulus, bulk modulus, modulus of rigidity	stress ÷ strain	Strain has no dimensions
Energy density	$u = \frac{E}{V}$	[ML²T⁻²] ÷ [L³]
Work per unit volume	$\frac{W}{V}$	Same as energy per volume
Radiation pressure	momentum flux on a surface	A real pressure exerted by light

Because energy per volume equals force per area, a gas pressure can be read as stored energy in every cubic metre. That link is why $PV$ has the units of work. See the stress and energy density pages for those cases, and Young's modulus for the elastic moduli. Pressure itself is a scalar, while stress in a solid can act along any face, but dimensions cannot tell the two apart.

Two dimensional analysis checks

Check: is the hydrostatic relation P = ρgh dimensionally correct?

Density: $[\rho] = [M L^{-3}]$ . Acceleration due to gravity: $[g] = [L T^{-2}]$ . Depth: $[h] = [L]$ .

Right side: $[M L^{-3}][L T^{-2}][L] = [M L^{-3+1+1} T^{-2}] = [M L^{-1} T^{-2}]$ .

Left side: [ML⁻¹T⁻²]. The powers agree, so the equation passes. A swimmer 2 m under water feels an extra 1000 × 9.8 × 2 = 1.96 × 10⁴ Pa, and the unit kg m⁻³ × m s⁻² × m does come out as Pa.

Convert 1 Pa into CGS units using the dimensions

In CGS, M is the gram, L the centimetre, T the second. Put the exponents of [M¹ L⁻¹ T⁻²] on the conversion factors.

$1\ \text{Pa} = (10^{3}\ \text{g})^{1}(10^{2}\ \text{cm})^{-1}(1\ \text{s})^{-2} = 10\ \text{g cm}^{-1}\text{s}^{-2}$

Answer: 1 Pa = 10 dyne/cm², or 1 dyne/cm² = 0.1 Pa. The minus one on L is what turns a factor of 100 into a division.

Mistakes that cost marks

Writing [ML²T⁻²]. That is energy. Pressure divides force by area; it does not multiply force by length.
Writing [MLT⁻²]. That is force. Students forget to divide by area at all when a question says "thrust".
Dropping the sign on L. [ML¹T⁻²] is a common slip. The area sits below the line, so the power of L must be negative.
Giving gauge and absolute pressure different dimensions. Both are pressures; the difference between them is also a pressure.
Treating atm or bar as base units. They are scaled pascals, so convert to Pa before any check.

Where pressure fits among its neighbors

Pressure sits between a few close relatives on the dimension chart. Comparing the brackets side by side shows how a single step of multiplying or dividing changes the answer.

Pressure next to related mechanics quantities
Quantity	Dimensional formula	Link to pressure
Force	[MLT⁻²]	pressure × area
Pressure	[ML⁻¹T⁻²]	force ÷ area
Energy or work	[ML²T⁻²]	pressure × volume
Density	[ML⁻³]	pressure ÷ (g × depth)
Surface tension	[MT⁻²]	pressure × length

Every row comes from the one before by a simple step, which is a good way to recall them. Learn the dimensional formula of work, then subtract three powers of L for pressure. The dimensional formulas hub lists the rest of the chart.

Frequently asked questions

What is the dimensional formula of pressure?

It is [M¹ L⁻¹ T⁻²]. Pressure is force per area, so [MLT⁻²] is divided by [L²], leaving one negative power of length.

What is the SI unit of pressure in base units?

The pascal. Writing the dimensions as base units gives kg m⁻¹ s⁻², which equals one newton per square metre.

Do pressure and stress have the same dimensions?

Yes. Both are force divided by area, so both are [ML⁻¹T⁻²] and both are measured in pascals. They differ in physical meaning, not in dimensions.

Why does energy density have the same dimensions as pressure?

Energy per volume is [ML²T⁻²] over [L³], which is [ML⁻¹T⁻²]. Multiply the top and bottom of force per area by a length and you get energy per volume, so the two always match.

Is pressure a vector or a scalar?

Pressure is a scalar. The force it produces on a surface has a direction, normal to that surface, but the pressure value has none.