Ring moment of inertia calculator
A thin ring of 3 kg and radius 4 m has a moment of inertia of 48 kg·m² about its centre, because all of its mass sits on the rim. Check a hoop with I = mr² without mixing it up with a disc. Free in your browser.
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Worked example
- Mass
- 3
- Radius
- 4
- Moment of inertia
- 48
The moment of inertia of a thin ring or hoop about its centre is one of the first closed forms students meet in rotational dynamics: every mass element sits at the same distance r from the central axis, so the integral of r squared dm collapses to I equals m times r squared. Homework, lab write-ups, and quick engineering checks need that identity without re-deriving polar inertia or confusing a thin hoop with a solid disc. This calculator accepts mass and radius in any consistent unit system, multiplies them as I equals m r squared, and returns the scalar moment of inertia together with the formula and a short axis description. Zero mass or zero radius correctly yields zero inertia.
Formula: I = m r² for a thin ring about its centre
A thin ring or hoop is the idealisation where all mass lies on a circle of radius r in a plane, with total mass m and negligible radial thickness. The mass moment of inertia about an axis is the integral of distance-squared times each mass element. When the axis passes through the geometric centre and is perpendicular to the plane of the ring, every element has the same perpendicular distance r, so I equals r squared times the integral of dm, which is simply m r squared.
The same result is often written I equals m R² when R denotes the hoop radius. Compare this carefully with a solid disc of the same outer radius, whose central-axis inertia is only one half m r squared because mass is distributed from the axis out to the rim. A thin spherical shell about a diameter uses two thirds m r squared, and a solid sphere uses two fifths—none of those shapes are this landing.
Units stay consistent: if m is in kilograms and r in metres, I is in kilogram metres squared; grams and centimetres, or slugs and feet, yield I in mass times length squared in the same system. The calculator does not convert units; it multiplies the numbers you send. Parallel-axis shifts, compound rings, and thick annuli are outside scope: reduce those problems to a thin hoop about its own centre first, or use a multi-shape inertia tool when the geometry is not a pure thin ring.
Who checks ring inertia with this tool
Classrooms and tutoring platforms use a dedicated ring moment of inertia calculator so learners can check I equals m r squared for a hoop without navigating a multi-shape form that defaults to a disc or rod. Exam generators and LMS plugins need a deterministic oracle that rejects a negative radius and never invents an answer when the student confused ring and disc factors.
Engineering notebooks that model flywheels approximated as thin rims, bicycle wheels as hoop mass, or ring-shaped space habitats often need only this closed form before coupling I into torque equals I alpha or rotational kinetic energy one half I omega squared. Failed validation does not consume balance. Prefer this capability when you already know the hoop is thin and the axis is the centre-perpendicular case; if your source gives diameter instead of radius, halve it first before calling.
What you can do with it
Homework checks for hoop I = m r²
Students enter mass and radius for a thin ring from textbook problems and compare the scalar I, formula, and axis note against a deterministic result before submitting.
Exam and LMS auto-grading
Mark ring-about-centre inertia items with a deterministic oracle that rejects negative radius and returns a stable scalar for batch scoring.
Simple rim and flywheel models
Supply thin-hoop inertia for wheel rims, ring-shaped links, and ideal flywheels in notebooks without a full solid-body CAD export.
FAQ
What is the moment of inertia of a thin ring about its centre?
For a thin ring or hoop of mass m and radius r, about an axis through the centre and perpendicular to the plane of the ring, I = m r².
How does a thin ring differ from a solid disc?
A thin ring has all mass at radius r, so I = m r². A solid disc of the same outer radius has I = (1/2) m r² about the same central axis because mass is distributed inward from the rim.
What happens if the radius is negative?
A negative radius is not physical for this model and is not billed. Negative mass is also rejected.
Which units should I use?
Any consistent pair: kilograms and metres yield kg·m².
Can I use it for a thick ring or an axis away from the centre?
No. It covers only a thin hoop turning about its own centre, with the axis perpendicular to its plane. For a thick annulus, a set of rings or a shifted axis, reduce the problem to a thin hoop first or use a multi-shape inertia tool.
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-ring \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"mass":3,"radius":4}'const res = await fetch("https://api.kit.forhosting.com/mech/moment-of-inertia-ring", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"mass": 3,
"radius": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/moment-of-inertia-ring",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"mass": 3,
"radius": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/moment-of-inertia-ring", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"mass":3,"radius":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"mass":3,"radius":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/moment-of-inertia-ring", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"mass": 3,
"radius": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.moment_of_inertia_ring",
"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.
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
A negative radius fails with invalid_input instead of a silent nonsense number; negative mass is likewise rejected.
Formula: I = m r² for a thin ring about its centre
That is the identity this endpoint evaluates: I equals m r².
Input contract, response fields, and errors
Negative mass is likewise rejected. Example: mass equals three kilograms and radius equals four metres gives I equals three times sixteen equals forty-eight kilogram metres squared. POST a JSON object with two required finite numbers: mass (m) and radius (r). Each field may be a JSON number or a numeric string; the aliases m and r are accepted when mass or radius are omitted. Missing fields, empty strings, NaN, and Infinity fail with invalid_input naming the field. Zero is allowed and returns moment_of_inertia equal to zero, which is mathematically correct for a zero-mass or zero-radius idealisation. On success the body echoes the cleaned mass and radius (rounded to twelve decimal places for stable goldens), the scalar moment_of_inertia, the formula string I = m r^2, a short axis description for the centre-perpendicular-to-plane axis, and a units object that reminds callers of kg·m² conventions. Pair the response with your own unit labels or diagrams if you need a sketch of the hoop and axis; this endpoint only computes the scalar inertia about the central axis and does not emit plots, multi-axis tensors, or unit conversions.
Where a dedicated ring inertia API earns its keep
Keep requests small, pin the rounded moment_of_inertia in tests, and you get reproducible I values for homework, lab write-ups, and edge workers without standing up symbolic math infrastructure of your own.
Technical questions
What does the ring moment of inertia calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.