Dimensional analysis checker
A newton and kg·m/s² both reduce to M L T⁻², so they match. Type the units of each side of a formula and see whether their seven SI base exponents agree, to catch a missing time factor or a homework identity that cannot hold. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Left
- N
- Right
- kg*m/s^2
- Summary
- dimensions match
- Left dimension
- M L T^-2
- Right dimension
- M L T^-2
- Left mass
- 1
Dimensional analysis is the first filter every physics and engineering equation should pass: both sides must share the same SI base dimensions, or the formula cannot be right. This checker accepts a left-hand unit product and a right-hand unit product, expands each symbol into the seven SI base exponents of mass, length, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, and reports whether those exponent vectors match. Use it to catch a missing factor of time, a force written where an energy belongs, or a homework identity that looks algebraic but is dimensionally impossible.
What dimensional consistency means in SI base units
Every mechanical, electrical, thermal, or optical quantity can be written as a product of the seven SI base dimensions: mass M, length L, time T, electric current I, thermodynamic temperature Θ, amount of substance N, and luminous intensity J. Force is M L T^-2, energy is M L^2 T^-2, pressure is M L^-1 T^-2, and a pure number, a radian, or a steradian is the dimensionless vector of all zeros.
An equation is dimensionally consistent only when the exponent vector of the left-hand side equals the exponent vector of the right-hand side, component by component. That test does not prove the equation is true — a missing factor of two still has the right dimensions — but it proves the equation cannot be true if the vectors differ.
Newton's second law in unit form is the classic example: the newton N expands to kg·m·s^-2, which is the same 7-vector as the product kg * m / s^2, so F = m a is dimensionally legal. Writing F = m v would pair M L T^-2 with M L T^-1 and fail immediately. You supply the units of each side.
SI prefixes never change the dimension, so kN and N match each other; a millijoule still has the dimensions of energy. When you need a converter, use a conversion capability; when you need to know whether an identity could possibly be right, use this checker.
Input contract, supported symbols, and how the parser reads a product
Each string is a unit product, not a full algebraic equation; do not put an equals sign in either field. Multiplication may be written with *, ·, ×, a textbook dot between symbols (kg.m.s-2), or juxtaposition with spaces (kg m / s^2). Division uses / .
Integer powers use ^, ** , Unicode superscripts, or a trailing integer glued to the symbol (m2 for square metres, s-2 for inverse seconds). Parentheses group factors so (kg*m)/s^2 is the same as kg*m/s^2. Numeric coefficients are dimensionless and ignored for the comparison, so 2*N has the same vector as N.
The catalogue includes the SI base units kg, g, m, s, A, K, mol, cd; the common derived units N, Pa, J, W, C, V, F, ohm, Ω, S, Wb, T, H, Hz, lm, lx, Bq, Gy, Sv, kat, rad, sr; extras such as min, h, L, eV, cal, bar, atm, torr, psi, dyn, erg, hp, lbf, ft, in, lb; SI prefixes from yocto to yotta including µ and u for micro; and full names such as newton, joule, watt, pascal, volt, ampere.
Exact table hits beat prefix splitting, so min is a minute of time rather than a milli-inch, and kg is the mass base rather than a prefix puzzle. Each side is capped at 400 characters and 64 tokens.
Where a quick dimension check pays off
Classrooms and tutoring platforms use a dimensional analysis calculator so learners can check Newton's law, the kinetic-energy identity, Ohm's law, or the ideal-gas law in unit form without expanding every derived symbol by hand. Exam generators need a deterministic oracle that marks F = m v wrong and F = m a right, and that rejects an invented unit instead of silently treating it as dimensionless.
Engineering notebooks and laboratory scripts often assemble a derived quantity from several sensors — a pressure from a force and an area, a power from a voltage and a current — and a one-call check of Pa versus N/m^2 or W versus J/s catches a swapped channel before the plot is published. Failed validation does not consume balance.
Keep each side short, pin the exponent integers in tests, and you get a reproducible consistency verdict for homework, CAD plugins, and edge workers without standing up a computer-algebra system of your own.
What you can do with it
Classroom and homework checks
Let students submit the units of each side of F = m a, K = ½ m v², or V = I R and verify that the SI exponent vectors match before they chase a numerical factor.
Exam and LMS auto-grading
Mark dimensional-analysis items with a deterministic oracle that accepts N versus kg*m/s^2, rejects J versus N, and fails unknown symbols instead of guessing.
Lab and notebook sanity checks
Confirm that a derived channel (Pa from N/m^2, W from J/s, V from W/A) is dimensionally legal before you log or plot it.
FAQ
What happens if a unit symbol is unknown?
The checker never invents a dimension for a token it does not recognise, and the failure is not billed.
Does a dimensional match prove the equation is true?
No. Matching exponents are necessary but not sufficient: a missing factor of two still has the right dimensions. A mismatch does prove the equation cannot be correct as written.
Which unit symbols can I use?
SI base and derived units (kg, m, s, A, K, mol, cd, N, Pa, J, W, V, ohm, Hz, …), common extras (min, L, eV, bar, ft, lbf), SI prefixes, and full names such as newton. Products use *, /, ^, juxtaposition, and parentheses.
Do prefixes such as kilo or milli change the check?
No. A prefix changes the size of a unit, never its dimension, so kN and N match each other and a millijoule still has the dimensions of energy.
Can I type a whole equation with an equals sign?
No. Enter the units of the left side and of the right side separately. The checker compares unit products, not variable names such as F or a, and it does not convert or solve anything.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/conv/dimensional-analysis \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"left":"N","right":"kg*m/s^2"}'const res = await fetch("https://api.kit.forhosting.com/conv/dimensional-analysis", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"left": "N",
"right": "kg*m/s^2"
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/conv/dimensional-analysis",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"left": "N",
"right": "kg*m/s^2"
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/conv/dimensional-analysis", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"left":"N","right":"kg*m/s^2"}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"left":"N","right":"kg*m/s^2"}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/conv/dimensional-analysis", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"left": "N",
"right": "kg*m/s^2"
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "conv.dimensional_analysis",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_chars | 400 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Unknown unit symbols fail with invalid_input instead of being guessed. The same pure parse logic runs free in the browser and at $0.002 per successful API call, so classroom checks and automation scripts never disagree on a Newton-versus-joule mix-up.
What dimensional consistency means in SI base units
This endpoint performs that comparison and nothing else: it does not convert magnitudes, it does not solve for a missing quantity, and it does not interpret variable names such as F or a.
Input contract, supported symbols, and how the parser reads a product
POST a JSON body with two required strings: left (aliases lhs, left_hand_side, lhs_units) and right (aliases rhs, right_hand_side, rhs_units). A missing field, an unmatched parenthesis, a non-integer exponent, or an unknown symbol such as foo fails with invalid_input naming the field; those failures are not billed. On success the body echoes both sides, a boolean consistent, a summary string, a compact left_dimension and right_dimension (for example M L T^-2, or 1 when dimensionless), and the seven left_* and right_* exponent integers so a client can render a table or diff without re-parsing.
Technical questions
How do I check if a physics equation is dimensionally consistent?
Send the units of each side as left and right, for example left=N and right=kg*m/s^2. The API expands both into SI base exponents and returns consistent true when every component matches.
What does the dimensional analysis checker cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.