Dimensional Analysis Calculator
Find the SI dimensions of a quantity, and check what multiplying or dividing two quantities gives.
Reference: dimensions of common quantities
| Quantity | SI unit | Dimensions | Base units |
|---|---|---|---|
| Length | m | L | m |
| Mass | kg | M | kg |
| Time | s | T | s |
| Electric current | A | I | A |
| Temperature | K | Θ | K |
| Amount of substance | mol | N | mol |
| Area | m² | L² | m² |
| Volume | m³ | L³ | m³ |
| Density | kg/m³ | M L⁻³ | kg·m⁻³ |
| Velocity | m/s | L T⁻¹ | m·s⁻¹ |
| Acceleration | m/s² | L T⁻² | m·s⁻² |
| Force | N | M L T⁻² | kg·m·s⁻² |
| Energy / work | J | M L² T⁻² | kg·m²·s⁻² |
| Power | W | M L² T⁻³ | kg·m²·s⁻³ |
| Pressure | Pa | M L⁻¹ T⁻² | kg·m⁻¹·s⁻² |
| Momentum | kg·m/s | M L T⁻¹ | kg·m·s⁻¹ |
| Frequency | Hz | T⁻¹ | s⁻¹ |
| Electric charge | C | T I | s·A |
| Voltage | V | M L² T⁻³ I⁻¹ | kg·m²·s⁻³·A⁻¹ |
| Resistance | Ω | M L² T⁻³ I⁻² | kg·m²·s⁻³·A⁻² |
| Capacitance | F | M⁻¹ L⁻² T⁴ I² | kg⁻¹·m⁻²·s⁴·A² |
| Inductance | H | M L² T⁻² I⁻² | kg·m²·s⁻²·A⁻² |
| Magnetic flux density | T | M T⁻² I⁻¹ | kg·s⁻²·A⁻¹ |
| Dynamic viscosity | Pa·s | M L⁻¹ T⁻¹ | kg·m⁻¹·s⁻¹ |
| Volumetric flow | m³/s | L³ T⁻¹ | m³·s⁻¹ |
| Entropy / heat capacity | J/K | M L² T⁻² Θ⁻¹ | kg·m²·s⁻²·K⁻¹ |
| Dimensionless ratio | 1 | 1 | 1 |
What dimensional analysis is for
Every physical quantity is built from seven SI base dimensions: mass (M), length (L), time (T), electric current (I), temperature (Θ), amount of substance (N) and luminous intensity (J). Writing a quantity in terms of those exponents strips away the choice of unit and leaves what the quantity is.
Force, for example, is M L T⁻² whether you measure it in
newtons, dynes or pounds-force.
The homogeneity check
Both sides of a valid equation must have identical dimensions. That single rule
catches a large share of algebra errors before you ever put numbers in — if one
side comes out as M L T⁻² and the other as
M L² T⁻², something is wrong, and no amount of
unit conversion will fix it.
Multiplying and dividing quantities adds and subtracts their exponents, which is
what the calculator above does. Force × length gives
M L² T⁻² — energy, as it should. Force ÷ area
gives M L⁻¹ T⁻² — pressure.
Dimensionless quantities
When every exponent cancels to zero the result is a pure number: strain, refractive index, Reynolds number, efficiency, Mach number. These are the quantities that mean the same thing in every unit system, which is exactly why they dominate engineering correlations.
What it cannot tell you
Dimensional analysis will not find a missing dimensionless constant. Kinetic
energy is ½mv², and the dimensions are identical whether
the factor is a half, a third, or 2π. It also cannot distinguish two different
quantities that share dimensions — torque and energy are both
M L² T⁻², yet they are not the same thing.