Dimension Formula
Before learning the dimension formula, let us recall what is dimension. Dimension in maths is a measure of the length, width, or height extended in a particular direction. By dimension definition, it is a measure of a point or line extended in one direction. Every shape around us has some dimensions. The concept of dimension in maths does not have any specific dimension formula. Dimension of any physical quantity is the power to which the fundamental units are raised to obtain one unit of that quantity. Let us learn about the dimension formula with a few examples in the end.
What Is the Dimension Formula?
The dimensional formula of any quantity is the expression showing the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity. If Q is any physical quantity, the expression representing its dimensional formula is given by:
Q = M^{a}L^{b}T^{c}
where,
 the exponents a, b and, c are called the dimensions.
The following table shows dimensional formulas for different physical quantities:
Physical quantity  Unit  Dimensional formula 
Length  m  L 
Mass  kg  M 
Time  s  T 
Acceleration or acceleration due to gravity  ms^{–2}  LT^{–2} 
Angle (arc/radius)  rad  M^{o}L^{o}T^{o} 
Angular displacement  rad  M^{o}l^{o}T^{o} 
Angular frequency (angular displacement/time)  rads^{–1}  T^{–1} 
Angular impulse (torque × time)  Nms  ML^{2}T^{–1} 
Angular momentum (Iω)  kgm^{2}s^{–1}  ML^{2}T^{–1} 
Angular velocity (angle/time)  rads^{–1}  T^{–1} 
Area (length × breadth)  m^{2}  L^{2} 
Boltzmann’s constant  JK^{–1}  ML^{2}T^{–2}θ^{–1} 
Bulk modulus (ΔP × (V/ΔV))  Nm^{–2}, Pa  M^{1}L^{–1}T^{–2} 
Calorific value  Jkg^{–1}  L^{2}T^{–2} 
Coefficient of linear or areal or volume expansion  ^{o}C^{–1} or K^{–1}  θ^{–1} 
Coefficient of surface tension (force/length)  Nm^{–1} or Jm^{–2}  MT^{–2} 
Coefficient of thermal conductivity  Wm^{–1}K^{–1}  MLT^{–3}θ^{–1} 
Coefficient of viscosity (F = η × A × (dv/dx))  poise  ML^{–1}T^{–1} 
Compressibility (1/bulk modulus)  Pa^{–1}, m^{2}N^{–2}  M^{–1}LT^{2} 
Density (mass / volume)  kgm^{–3}  ML^{–3} 
Displacement, wavelength, focal length  m  L 
Electric capacitance (charge/potential)  CV^{–1}, farad  M^{–1}L^{–2}T^{4}I^{2} 
Electric conductance (1/resistance)  Ohm^{–1} or mho or siemen  M^{–1}L^{–2}T^{3}I^{2} 
Electric conductivity (1/resistivity)  siemen/metre or Sm^{–1}  M^{–1}L^{–3}T^{3}I^{2} 
Electric charge or quantity of electric charge (current × time)  coulomb  IT 
Electric current  ampere  I 
Electric dipole moment (charge × distance)  Cm  LTI 
Electric field strength or Intensity of electric field (force/charge)  NC^{–1}, Vm^{–1}  MLT^{–3}I^{–1} 
Electric resistance (potential difference/current)  ohm  ML^{2}T^{–3}I^{–2} 
Emf (or) electric potential (work/charge)  volt  ML^{2}T^{–3}I^{–1} 
Energy (capacity to do work)  joule  ML^{2}T^{–2} 
Energy density (energy/volume)  Jm^{–3}  ML^{–1}T^{–2} 
Entropy (ΔS = ΔQ/T)  Jθ^{–1}  ML^{2}T^{–2}θ^{–1} 
Force (mass x acceleration)  newton (N)  MLT^{–2} 
Force constant or spring constant (force/extension)  Nm^{–1}  MT^{–2} 
Frequency (1/period)  Hz  T^{–1} 
Gravitational potential (work/mass)  Jkg^{–1}  L^{2}T^{–2} 
Heat (energy)  J or calorie  ML^{2}T^{–2} 
Illumination (Illuminance)  lux (lumen/metre^{2})  MT^{–3} 
Impulse (force x time)  Ns or kgms^{–1}  MLT^{–1} 
Inductance (L) (energy = \(\frac{1}{2}\) LI^{2} or
Coefficient of selfinduction 
henry (H)  ML^{2}T^{–2}I^{–2} 
Intensity of gravitational field (F/m)  Nkg^{–1}  L^{1}T^{–2} 
Intensity of magnetization (I)  Am^{–1}  L^{–1}I 
Joule’s constant or mechanical equivalent of heat  Jcal^{–1}  M^{o}L^{o}T^{o} 
Latent heat (Q = mL)  Jkg^{–1}  M^{o}L^{2}T^{–2} 
Linear density (mass per unit length)  kgm^{–1}  ML^{–1} 
Luminous flux  lumen or (Js^{–1})  ML^{2}T^{–3} 
Magnetic dipole moment  Am^{2}  L^{2}I 
Magnetic flux (magnetic induction x area)  weber (Wb)  ML^{2}T^{–2}I^{–1} 
Magnetic induction (F = Bil)  NI^{–1}m^{–1} or T  MT^{–2}I^{–1} 
Magnetic pole strength  Am (ampere–meter)  LI 
Modulus of elasticity (stress/strain)  Nm^{–2}, Pa  ML^{–1}T^{–2} 
Moment of inertia (mass × radius^{2})  kgm^{2}  ML^{2} 
Momentum (mass × velocity)  kgms^{–1}  MLT^{–1} 
Permeability of free space \(\left(μ_o = \dfrac{4\pi Fd^{2}}{m_1m_2}\right)\)  Hm^{–1} or NA^{–2}  MLT^{–2}I^{–2} 
Permittivity of free space \(\left({{\varepsilon }_{o}}=\frac{{{Q}_{1}}{{Q}_{2}}}{4\pi F{{d}^{2}}}\right)\)  Fm^{–1} or C^{2}N^{–1}m^{–2}  M^{–1}L^{–3}T^{4}I^{2} 
Planck’s constant (energy/frequency)  Js  ML^{2}T^{–1} 
Poisson’s ratio (lateral strain/longitudinal strain)  ––  M^{o}L^{o}T^{o} 
Power (work/time)  Js^{–1} or watt (W)  ML^{2}T^{–3} 
Pressure (force/area)  Nm^{–2} or Pa  ML^{–1}T^{–2} 
Pressure coefficient or volume coefficient  ^{o}C^{–1} or θ^{–1}  θ^{–1} 
Pressure head  m  M^{o}LT^{o} 
Radioactivity  disintegrations per second  M^{o}L^{o}T^{–1} 
Ratio of specific heats  ––  M^{o}L^{o}T^{o} 
Refractive index  ––  M^{o}L^{o}T^{o} 
Resistivity or specific resistance  Ω–m  ML^{3}T^{–3}I^{–2} 
Specific conductance or conductivity (1/specific resistance)  siemen/metre or Sm^{–1}  M^{–1}L^{–3}T^{3}I^{2} 
Specific entropy (1/entropy)  KJ^{–1}  M^{–1}L^{–2}T^{2}θ 
Specific gravity (density of the substance/density of water)  ––  M^{o}L^{o}T^{o} 
Specific heat (Q = mst)  Jkg^{–1}θ^{–1}  M^{o}L^{2}T^{–2}θ^{–1} 
Specific volume (1/density)  m^{3}kg^{–1}  M^{–1}L^{3} 
Speed (distance/time)  ms^{–1}  LT^{–1} 
Stefan’s constant \(\left( \frac{\text{heat energy}}{\text{area} \times \text{time} \times \text{temperature}^{4}} \right)\)  Wm^{–2}θ^{–4}  ML^{o}T^{–3}θ^{–4} 
Strain (change in dimension/original dimension)  ––  M^{o}L^{o}T^{o} 
Stress (restoring force/area)  Nm^{–2} or Pa  ML^{–1}T^{–2} 
Surface energy density (energy/area)  Jm^{–2}  MT^{–2} 
Temperature  ^{o}C or θ  M^{o}L^{o}T^{o}θ 
Temperature gradient \(\left(\frac{\text{change in temperature}}{\text{distance}}\right)\)  ^{o}Cm^{–1} or θm^{–1}  M^{o}L^{–1}T^{o}θ 
Thermal capacity (mass × specific heat)  Jθ^{–1}  ML^{2}T^{–2}θ^{–1} 
Time period  second  T 
Torque or moment of force (force × distance)  Nm  ML^{2}T^{–2} 
Universal gas constant (work/temperature)  Jmol^{–1}θ^{–1}  ML^{2}T^{–2}θ^{–1} 
Universal gravitational constant \(\left(F = G. \frac{{{m}_{1}}{{m}_{2}}}{{{d}^{2}}}\right)\)  Nm^{2}kg^{–2}  M^{–1}L^{3}T^{–2} 
Velocity (displacement/time)  ms^{–1}  LT^{–1} 
Velocity gradient (dv/dx)  s^{–1}  T^{–1} 
Volume (length × breadth × height)  m^{3}  L^{3} 
Water equivalent  kg  ML^{o}T^{o} 
Work (force × displacement)  J  ML^{2}T^{–2} 
You can see the applications of the dimension formulas in the section below.
Solved Examples Using Dimension Formulas

Example 1: Using dimension formula, Q = M^{a}L^{b}T^{c}, find the values of a, b, and c if the given quantity is velocity.
Solution:
To find: Values for a, b, and c
Given:
Quantity = Velocity
Using the dimension formula,
Q = M^{a}L^{b}T^{c}
We know,
Velocity = (displacement/time)
= L/T
= M^{0}L^{1}T^{1}
Comparing with dimension formula, we get,
a = 0, b = 1, c = 1
Answer: a = 0, b = 1, c = 1

Example 2: Find the dimension formula of momentum.
Solution:
To find: Dimension formula of momentum
We know,
Momentum = (mass × velocity)
= [MLT^{1}]
Answer: Dimension formula for momentum = [MLT^{1}]