Dimensional Analysis Checker
Check that an equation’s units balance. Declare each variable’s unit, type the equation, and see the dimension of every term in mass, length, time, current, temperature, amount and luminous intensity, with the term that does not fit flagged. Or find the SI unit of any expression.
How to Use
- Choose Check an equation, or SI unit of an expression to find what one expression is measured in.
- Declare each variable on its own line with its unit in brackets: v [m/s], a [m/s^2], R [J/(mol*K)]. Leave the brackets empty for a pure number.
- Type the equation, such as s = u*t + 1/2*a*t^2. Use * or · between factors, ^ or ² ³ ⁻¹ for powers, and brackets where needed. Names you did not declare are read as units (N, J, kW, h, eV …).
- Read the verdict. The table shows the exponents of M, L, T, I, Θ, N and J for every term; a term that does not match the left side turns red and is named in the message.
- Open Show Work for each variable’s dimension, each term’s dimension and SI unit, and a note when the dimensions agree but the unit sizes do not (km/h with seconds and metres).
Worked Example
A consistent equation. In s = ut + ½at², declare s [m], u [m/s], a [m/s²] and t [s]. The left side is L. The first term on the right is (L·T⁻¹) × T = L. The second is ½ × (L·T⁻²) × T² = L, because the ½ is a pure number. Every term is L, measured in metres: consistent.
An inconsistent one. Someone writes s = v/t. The right side is (L·T⁻¹) ÷ T = L·T⁻², the dimension of an acceleration, while the left side is L. No choice of numbers can make a distance equal an acceleration, so the formula is wrong (it should be s = v·t).
The common mistake: trusting the check with mixed units. With v declared in km/h, t in s and s in m, s = v·t is dimensionally fine, but 1 km/h × 1 s is only 0.2778 m. A car at 72 km/h for 10 s covers 72 ÷ 3.6 × 10 = 200 m, not the 720 you get by multiplying the raw numbers. The checker says when the declared units differ in size like this, so convert before calculating.
Show Work
Formulas
From Fourier to the Buckingham π Theorem
Joseph Fourier set out the rule that both sides of a physical equation must have the same dimensions in Théorie analytique de la chaleur (1822), where he gave each quantity in his heat equations an exponent of length, time and temperature. James Clerk Maxwell popularised the notation of writing dimensions as powers of mass, length and time, [M], [L] and [T], in the 1870s.
Lord Rayleigh made the method a working tool for guessing how physical quantities depend on each other, and in 1914 Edgar Buckingham published the theorem now named after him: a relation among n quantities built from k base dimensions can be rewritten in terms of n − k dimensionless groups, such as the Reynolds and Mach numbers.
Getting dimensions right is not the same as getting units right. NASA’s Mars Climate Orbiter was lost in September 1999 because one team’s software reported thruster impulse in pound-force seconds while the navigation software expected newton-seconds: the same dimension, but a factor of 4.45 apart. Today the SI has seven base units, and since the 2019 redefinition all of them are fixed by exact values of physical constants.
About This Tool
This checker reads units and equations with its own parser. It handles *, ·, ×, /, brackets, ^ powers including fractions such as ^(1/2), superscripts like m² and s⁻¹, units written side by side (kg m s^-2), every SI prefix from quecto to quetta, the named SI derived units, and common non-SI units such as hours, litres, electronvolts, bar, feet and pounds. Each quantity becomes a list of exact rational exponents of the seven base dimensions. Every term on each side of the equation is compared with the left side, inner sums and function arguments are checked too, and a note appears when the dimensions agree but the unit sizes do not. A declared variable name takes priority over a unit symbol, so V can be a volume in one equation and volts in another.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Unit Converter, Error Propagation Calculator, and Physical Constants Table.
Frequently Asked Questions
What does dimensional analysis check?
That every term added or equated has the same dimension. In s = ut + ½at², s is L (length), u·t is (L·T⁻¹)(T) = L and ½·a·t² is (L·T⁻²)(T²) = L, so the equation is consistent. In s = v/t the right side is (L·T⁻¹)/T = L·T⁻², an acceleration, so it cannot equal a distance.
Can an equation pass the check and still be wrong?
Yes. Pure numbers have no dimension, so s = ut + at² passes even though the ½ is missing. Unit sizes can also differ: with v in km/h and t in seconds, v·t has the dimension of length but 72 km/h for 10 s is 200 m, not 720. The checker flags the 0.2778 size mismatch.
What are the seven base dimensions?
Mass M (kilogram), length L (metre), time T (second), electric current I (ampere), temperature Θ (kelvin), amount of substance N (mole) and luminous intensity J (candela). Every SI unit is a product of their powers: a joule is kg·m²·s⁻², so energy is M·L²·T⁻²; a volt is kg·m²·s⁻³·A⁻¹.
How do I type units such as J/(mol·K)?
Write J/(mol*K), J/(mol·K) or J mol^-1 K^-1. Without the brackets, J/mol K is read left to right as (J/mol)·K and the checker warns you. Superscripts work too: kg·m²·s⁻² is the joule, N/m2 is read as N/m² (1 Pa), and 1 kW·h = 3,600,000 J.
Why must the inside of sin, exp or ln have no units?
Because e^x = 1 + x + x²/2 + …, and 1, x and x² can only be added if x is a pure number. So sin(ωt) is fine (ω in s⁻¹ times t in s), but sin(x) with x in metres is flagged: the argument has dimension L.
How do I use the Dimensional Analysis Checker?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Checking homework
Before using s = v/t in an answer, the checker shows the right side is L·T⁻² against L on the left, so the formula is wrong before any numbers go in.
Mixed unit systems
With v in km/h, t in s and s in m, s = v·t passes dimensionally but carries a factor of 0.2778: 72 km/h for 10 s is 200 m, not 720.
Chemistry and gases
PV = nRT with P in Pa, V in m³ and R in J/(mol·K) gives M·L²·T⁻² (joules) on both sides.
Energy bills
kW·h works out as a joule times 3,600,000, so a 1.5 kW heater run for 2 h uses 3 kW·h, or 10.8 MJ.
Dimensionless groups
The Reynolds number ρvL/μ, with ρ in kg/m³, v in m/s, L in m and μ in Pa·s, comes out dimensionless, as a Buckingham π group must.
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