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Density Dimensional Formula

Dimensional formulas of physical quantitiesClass 9Class 11Updated 1 Nov 2026

[ρ]=[M1L−3T0][\rho] = [M^{1} L^{-3} T^{0}]
SI unit
kg/m³
Dimensions
[M¹ L⁻³ T⁰]
where
  • M dimension of mass kg
  • L dimension of length m
  • T dimension of time (power 0) s

The density dimensional formula is [M¹ L⁻³ T⁰], usually shortened to [ML⁻³]. Density depends on mass to the first power and on length to the power minus three, and it does not depend on time at all.

Density is mass packed into each unit of volume, ρ = m/V. Mass is a base quantity and volume is a length cubed, so the minus three on L comes straight from the cube. Below are the derivation, the unit kg/m³, how density differs from its linear and surface cousins, a sound speed check and the slips examiners see most. For calculations with real materials, use the density formula page.

Quick facts

  • Dimensional formula

    [M¹ L⁻³ T⁰], short form [ML⁻³]

  • Defining formula

    ρ=mV\rho = \frac{m}{V}

  • SI unit

    kg/m³

  • Other unit

    g/cm³, where 1 g/cm³ = 1000 kg/m³

  • Scalar or vector

    Scalar

  • Common symbol

    ρ (rho)

Working out the density dimensional formula

Only two base quantities appear, so the working is short.

  1. Definition: ρ=mV\rho = \frac{m}{V}.
  2. Mass: [m]=[M][m] = [M].
  3. Volume of a cube of side ll is l3l^{3}, so [V]=[L3][V] = [L^{3}].
  4. Divide: [ρ]=[M]÷[L3]=[ML−3][\rho] = [M] \div [L^{3}] = [M L^{-3}]. With time shown, [M1L−3T0][M^{1} L^{-3} T^{0}].

Writing T⁰ is optional. Some Class 11 answer keys show it to make clear that time was checked and is absent; others leave it out. Both forms earn the mark.

The SI unit and unit conversion

Swap M for kilogram and L⁻³ for per cubic metre. The SI unit of density is kg m⁻³, written kg/m³. There is no special name.

Density units and their size in SI
UnitValue in kg/m³Typical use
g/cm³1000Lab samples, chemistry
kg/L1000Liquids
g/L1Gases

Convert the density of iron, 7.87 g/cm³, to SI units

Use the powers in [M L⁻³]: 7.87×10−3 kg(10−2 m)3=7.87×10−310−6 kg/m3\frac{7.87 \times 10^{-3}\ \text{kg}}{(10^{-2}\ \text{m})^{3}} = \frac{7.87 \times 10^{-3}}{10^{-6}}\ \text{kg/m}^3.

Result: 7.87 × 10³ kg/m³, or 7870 kg/m³. The cube on the length factor is why the answer jumps by a thousand, not a hundred.

Quantities with the same bracket, and ones that only look alike

Few common quantities share [ML⁻³] exactly, because "per volume" is the defining idea. The ones that do are all mass per volume in some form.

Mass per volume quantities, all [ML⁻³]
QuantityWhere you meet it
Density of a solid or liquidFloating, sinking, buoyancy
Density of a gasIdeal gas, ρ = PM/(RT)
Mass concentration of a solutionkg of solute per m³

The nearby densities are worth separating, since they look similar in words but differ in dimensions:

Volume, surface and linear density compared
Kind of densityDefinitionDimensions
Volume density ρmass ÷ volume[ML⁻³]
Surface density σmass ÷ area[ML⁻²]
Linear density λmass ÷ length[ML⁻¹]
Relative densityρ ÷ ρ of waterdimensionless

Relative density, also called specific gravity, is a ratio of two densities, so every power cancels. See relative density for that case and linear density for strings.

A dimensional check using density

Check: is the speed of sound v = √(B/ρ) dimensionally correct?

Bulk modulus is a pressure: [B]=[ML−1T−2][B] = [M L^{-1} T^{-2}]. Density: [ρ]=[ML−3][\rho] = [M L^{-3}].

Ratio: [ML−1T−2][ML−3]=[L2T−2]\frac{[M L^{-1} T^{-2}]}{[M L^{-3}]} = [L^{2} T^{-2}]. Its square root is [LT−1][L T^{-1}].

That is a speed, so the equation is consistent. Notice how the mass cancels: density is what lets a pressure turn into a speed.

The same pairing of density with pressure appears in P = ρgh, which the pressure page checks.

Errors to avoid

  • Writing [ML³]. Volume is in the denominator, which makes the length exponent −3.
  • Using [ML⁻²] for ordinary density. That is mass per area, the surface density of a sheet.
  • Giving relative density the unit kg/m³. It is a pure number with no unit.
  • Forgetting to cube the conversion factor. A cubic centimetre is one millionth of a cubic metre.
Quick memory hook: density has a T⁰ because a block of wood has the same density whether it is still or moving.

Where density shows up next

Density links mass to volume, so it turns up wherever both appear. In fluids it sets pressure at depth and buoyant force. In waves it sets the speed of sound. In gravitation, a planet's mean density multiplies its volume to give its mass.

The bracket [ML⁻³] stays fixed in all of these. If an equation you have written puts density next to a pressure, expect the leftover to be [L²T⁻²], a speed squared, or a length times g. Every other bracket in the chapter is on the dimensional formulas page.

Frequently asked questions

Which SI unit is used for density?

Kilogram per cubic metre, kg/m³. It follows from [ML⁻³] by reading M as kg and L as m.

What is the dimensional formula for density?

Density works out to [M¹ L⁻³ T⁰], which is mass divided by length cubed. Time does not appear.

Is density a scalar or a vector?

Scalar. A density value tells you how much mass sits in each cubic metre, with no direction attached.

Why is the power of time zero in density?

Neither mass nor volume involves time, so dividing one by the other leaves no T term. Writing T⁰ just records that.

How is density different from relative density in dimensions?

Density is [ML⁻³] with unit kg/m³. Relative density divides one density by another, so it is dimensionless and has no unit.