Moment of inertia calculator
A solid disc of 2 kg and 3 m radius has a moment of inertia of 9 kg·m² about its central axis. Pick ring, disc, sphere or rod, enter the mass and the radius or length, and get I with the matching textbook formula. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Rigid-body shape: ring
- Disc
- Mass
- 2
- Resistance (Ω)
- 3
- Mass
- 2
- I
- 9
- Moment of inertia
- 9
- Inertia
- 9
A moment of inertia calculator turns the standard rigid-body formulae into a practical tool for homework, lab checks, and quick engineering estimates. Choose a common shape—thin ring (hoop), solid disc, solid sphere, or thin rod—enter mass and the matching radius or rod length in a consistent unit system, and the tool returns the scalar moment of inertia I about the usual centre-of-mass axis, together with the textbook formula, a fully substituted equation, a short physics message, and ordered step strings you can paste into worksheets or automated tests. Zero mass yields zero inertia when dimensions are valid.
What moment of inertia means and which shapes this tool covers
Moment of inertia measures how a rigid body resists angular acceleration about a chosen axis, playing the same role for rotation that mass plays for translation. For a continuous body it is the integral of distance-squared times mass element; for classroom and design work the result collapses to compact closed forms when the body is a symmetric idealisation. This calculator implements four of the most common centre-of-mass cases students and engineers look up online.
A thin ring or hoop rotating about the axis through its centre and perpendicular to its plane has all mass at radius r, so I equals m times r squared. A solid disc or cylinder of uniform density about its central symmetry axis has I equal to one-half m r squared. A solid sphere about a diameter through its centre has I equal to two-fifths m r squared.
A thin rod of length L about an axis through its midpoint and perpendicular to its length has I equal to one-twelfth m L squared. Mass must be non-negative; negative mass is physically rejected. Radius and length must also be non-negative because the formulae use squared dimensions.
Keep m, r, and L in one coherent unit system—for example kilograms and metres yield I in kilogram metres squared. The tool does not apply the parallel-axis theorem, does not integrate arbitrary CAD meshes, and does not convert units between SI and imperial: reduce or convert those problems yourself before calling the solver. Hollow shells, thick-walled cylinders, and rods about an end are outside this landing’s shape list on purpose so search intent stays aligned with the four standard textbook forms.
Practical tips, limits, and related searches
Keep mass and lengths inside the published absolute range so products and rounding stay stable. Use one consistent unit system end to end before calling; this landing does not perform unit conversion. When teaching rotational dynamics, treat this tool as the I lookup step only: combine I with angular acceleration to get torque, or with omega for rotational kinetic energy one-half I omega squared in a separate step.
If your problem needs I about a parallel axis a distance d from the centre of mass, apply the parallel-axis theorem I_parallel equals I_cm plus m d squared yourself after reading I_cm from this calculator. This page targets people searching for a moment of inertia calculator, hoop or disc I equals m r squared solver, solid sphere two-fifths m r squared, or thin rod one-twelfth m L squared online; it is not a full multibody dynamics suite, not a CAD inertia tensor exporter, and not a unit converter.
Prefer this capability when you already know mass and the characteristic radius or rod length for one of the four listed shapes; if your source gives diameter instead of radius, halve it first. Round-trip checks are easy: recompute k times m times dim squared by hand and confirm it equals the published I within the same twelve-decimal rounding used in the response.
What you can do with it
Homework checks for standard I formulae
Students enter mass and radius or length for a ring, disc, sphere, or rod from textbook problems and compare the scalar I, formula, and substitution steps against a deterministic result before submitting.
Lab pre-check before parallel-axis or energy work
Pipelines validate centre-of-mass I from measured mass and geometry first, then apply parallel-axis shifts or rotational kinetic energy only when the base I matches lab notes.
FAQ
What formulae does the moment of inertia calculator use?
Ring (hoop): I = m r^2. Disc: I = (1/2) m r^2. Solid sphere: I = (2/5) m r^2. Thin rod about centre: I = (1/12) m L^2. All are about the standard centre-of-mass axis described in the response.
Can mass be negative?
Classical moment of inertia requires m ≥ 0.
What length do I enter for a rod?
Use field L (or length) for the full rod length. If L is omitted, the tool accepts the rod length in the r field so single-length forms still work.
In what units is the result?
When mass is in kilograms and radius or length in metres, I is in kg·m². Keep all inputs in one consistent unit system; the calculator does not convert units.
Is the moment of inertia calculator free?
The browser tool is free and runs locally.
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/mech/moment-of-inertia \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"shape":"disc","m":2}'const res = await fetch("https://api.kit.forhosting.com/mech/moment-of-inertia", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"shape": "disc",
"m": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/moment-of-inertia",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"shape": "disc",
"m": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/moment-of-inertia", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"shape":"disc","m":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"shape":"disc","m":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/moment-of-inertia", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"shape": "disc",
"m": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.moment_of_inertia",
"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_abs | 1000000000000 |
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
Negative mass is rejected with a clear invalid_input error so incomplete or unphysical forms never silently invent a number. Pricing is a flat $0.002 per successful API request; failed validation is never charged.
How to call the API and what comes back
Send a JSON body with required fields shape and m, plus the length scale for that shape. Shape accepts ring, disc, sphere, or rod, with friendly aliases such as hoop, disk, solid_sphere, thin_rod, and bar. Mass m (alias mass) is a finite non-negative number within the published absolute limit of one trillion. For ring, disc, and sphere provide radius r (aliases radius and R). For rod provide length L (aliases length and l), or pass the rod length in the r field when L is omitted so single-field forms still work. Absolute values must stay within the limit so products remain finite in IEEE double precision on the Worker and in the browser. Failed validation—missing shape, unknown shape token, missing mass or length scale, non-finite numbers, values outside absolute limits, negative mass, or negative radius or length—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price. Prefer JSON numbers when you control the client; string numbers are accepted only when they match a strict finite numeric pattern for homework-form paste paths.
Practical tips, limits, and related searches
Pair classroom examples deliberately: mass two and radius three for a disc yields I equals nine; mass five and radius two for a solid sphere yields I equals eight; mass three and radius four for a ring yields I equals forty-eight; mass twelve and length six for a rod yields I equals thirty-six; and a negative mass demonstrates the hard invalid_input path. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately try negative mass.
Integration use cases
Unit tests for sim and game rotation helpers
Generate known-good moment-of-inertia cases plus deliberate negative-mass rejections for coding kata expected values in physics engines and rigid-body modules.