Dimensional analysis checker: does your equation balance in SI units?

What it does

Every correct physics equation is dimensionally consistent: both sides, and every term added together, have the same dimensions. You cannot add a length to a time, and the argument of exp, ln or sin must be a pure number. Checking this is the quickest way to catch an algebra slip, a missing square or a wrong formula from memory.

This checker turns each symbol’s unit into powers of the seven SI base units (kg, m, s, A, K, mol, cd), works through your equation, and tells you whether it balances. If it does not, it points to the place: two terms that cannot be added, a dimensioned argument to a function, or two sides that differ.

How to use it

1. Type the equation with = between the sides, for example E = 1/2*m*v^2. Use * for multiplication, / for division and ^ for powers. A chain like E = F*x = m*v^2 checks all three; a formula without = just shows its dimensions.
2. List the unit of each symbol, one per line: m = kg, v = m/s, rho = kg/m^3, C_d = 1 for a pure number. Units can use prefixes (km, µF, MPa), named units (N, J, W, Pa, V, Ω, T, eV, L, atm, kWh) and forms like kg·m·s⁻² or J/(mol K).
3. Leave common constants ticked to use c, g, G, h, hbar, k_B, e, epsilon_0, mu_0, N_A, R and sigma without declaring them. Any unit you give a name yourself wins.
4. Read the verdict, the list of problems, and the dimensions of each side and each added term. Download the result as an image, PDF or Word file.

The last box converts any unit to SI base units: V/m is kg m s⁻³ A⁻¹, Ω m is kg m³ s⁻³ A⁻², kWh is a joule.

Worked example

The page opens with E = 1/2·m·v² and E in J, m in kg, v in m/s. Both sides come out as kg m² s⁻² (the joule), so it balances. Pick the drag force example, written by mistake as F = ½ ρ v C_d A: the left side is kg m s⁻² (N) but the right side is kg s⁻¹, so it does not balance; the correct formula has v². The pendulum example T = 2π√(L/g) uses the built-in g and balances in seconds, and the ideal gas law pV = nRT balances in joules.

Tips

• Dimensional analysis cannot see pure-number factors: E = m v² balances even though the ½ is missing.
• Angles (rad) and counts are dimensionless, so ω in rad/s works like 1/s.
• Temperatures in °C are treated as differences (the same dimension as K).

FAQ

Why does it say a power must be a fixed number? x^n has a definite unit only when n is a number such as 2 or ½. A variable exponent is fine when the base is dimensionless.

Can it convert between units? It checks dimensions only; scale factors (such as km to m) do not change whether an equation balances.

Last updated 2026-11-09.