Hollow sphere moment of inertia
A thin spherical shell of 3 kg and radius 2 m has a moment of inertia of 8 kg·m² about any diameter, from I = (2/3) mr². That factor is larger than a solid ball's, and every step is shown. Free in your browser.
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Worked example
- M
- 3
- Resistance (Ω)
- 2
- Mass
- 3
- Radius
- 2
- R
- 2
- I
- 8
A hollow sphere moment of inertia calculator answers a search that solid-ball and multi-shape tools often bury: the thin spherical shell, where all mass sits on a surface of radius r, has I equal to two thirds m r squared about any diameter through the centre. Students checking textbook tables, engineers sizing spherical pressure vessels that rotate about an axis, and simulation authors validating rigid-body inertia tensors need that exact factor—not the solid-sphere two-fifths factor—without picking a shape enum first. Enter non-negative mass m and non-negative radius r in a consistent unit system; the tool returns I in kg·m² when m is in kilograms and r in metres, plus formula, substituted equation, axis description, and ordered steps.
What I = (2/3) m r² means for a thin spherical shell
For a thin spherical shell—all mass on a sphere of radius r with negligible wall thickness—every mass element is at geometric distance r from the centre, but the perpendicular distance to a chosen diameter axis varies with latitude on the sphere. Integrating over the uniform shell yields the closed form I equals two thirds times m times r squared about any diameter through the centre.
That factor two thirds is larger than the solid uniform sphere factor two fifths, which is intuitive: a shell places more mass farther from the axis on average than a filled ball of the same outer radius. The result is independent of which diameter you choose by spherical symmetry. This calculator implements only that shell formula; it does not integrate thick-walled hollow spheres with inner and outer radii, does not treat spherical shells about a tangent axis, and does not apply the parallel-axis theorem for an offset spin axis.
If your problem is a solid ball, use a solid-sphere path with I equals two fifths m r squared instead. If your problem is a thin hoop in a plane, I equals m r squared about the central axis perpendicular to the plane—again a different factor.
Who needs the thin-shell formula
Classrooms and tutoring platforms use a dedicated hollow sphere moment of inertia calculator so learners can check the shell row of inertia tables without confusing it with the solid sphere or thin hoop rows. Exam-prep generators need a deterministic oracle that marks negative radius invalid instead of returning a positive I that a buggy grader might accept. Failed validation does not consume balance.
Keep requests small, pin the rounded I in tests, and you get reproducible shell inertia for homework, LMS plugins, and edge workers without standing up a general CAD inertia package of your own. Prefer this capability when you already know you need the thin-shell diameter formula; use a multi-shape moment of inertia tool when the body type is still a dropdown among ring, disc, solid sphere, and rod.
What you can do with it
Homework checks for thin-shell tables
Students enter mass and radius from textbook spherical-shell problems and compare I = (2/3) m r² and substitution steps against a deterministic result before submitting.
Pre-check for thin-walled rotating vessels
Pipelines validate diameter-axis shell inertia from design mass and radius first, then branch to thick-wall or offset-axis models only when the thin-shell check matches notes.
FAQ
What is the moment of inertia of a hollow sphere about a diameter?
For a thin spherical shell of mass m and radius r, I = (2/3) m r² about any diameter through the centre.
How is a hollow sphere different from a solid sphere?
A solid uniform sphere uses I = (2/5) m r² about a diameter. A thin hollow shell uses I = (2/3) m r². This calculator implements only the shell (hollow) formula.
What happens if the radius is negative?
Radius must be non-negative because I = (2/3) m r² is defined for a physical shell radius r ≥ 0. Invalid input is not billed.
Which units does the hollow sphere moment of inertia calculator use?
Any consistent system. With m in kilograms and r in metres, I is in kg·m². The tool multiplies the numbers you send and does not convert units.
Does it work for a thick-walled hollow sphere?
No. It assumes a wall so thin that all the mass sits at one radius. A sphere with distinct inner and outer radii, a spin axis tangent to the surface or an offset axis needs a different formula.
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-sphere-hollow \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"m":3,"r":2}'const res = await fetch("https://api.kit.forhosting.com/mech/moment-of-inertia-sphere-hollow", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"m": 3,
"r": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/moment-of-inertia-sphere-hollow",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"m": 3,
"r": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/moment-of-inertia-sphere-hollow", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"m":3,"r":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"m":3,"r":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/moment-of-inertia-sphere-hollow", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"m": 3,
"r": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.moment_of_inertia_sphere_hollow",
"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 rather than silently squaring a nonsense length. Pricing is a flat $0.002 per successful API request; failed validation is never charged.
What I = (2/3) m r² means for a thin spherical shell
Classical rigid-body mechanics defines the moment of inertia of a mass distribution about an axis as the integral of r_perp squared dm. Keep mass non-negative and radius non-negative; classical inertia tables assume physical length and mass scales, and this endpoint rejects a negative radius with invalid_input so homework forms cannot square a signed length by accident and still look correct.
Input contract, response shape, and error behaviour
Negative mass is likewise rejected. Example: m equals three and r equals two yields I equals eight, because two thirds times three times four equals eight. POST a JSON body with two required finite numbers: m (alias mass), the non-negative mass of the shell, and r (aliases radius and R), the non-negative outer radius of the thin shell. Each field may arrive as a JSON number or as a numeric string matching a strict finite pattern; missing fields, empty strings, non-finite values, and absolute magnitudes beyond one trillion fail with invalid_input naming the field. Pair the response with your own unit conversion if inputs are not SI; this endpoint multiplies the numbers you send and does not convert grams to kilograms or centimetres to metres.
Where a hollow-sphere inertia API earns its keep
Engineering notebooks for spherical tanks, satellite fuel shells, and thin-walled rotating vessels often need only the diameter-axis shell formula before applying parallel-axis shifts elsewhere; calling a tiny parse API avoids pulling a full multi-shape inertia suite for a two-parameter problem.
Integration use cases
Unit tests for rigid-body inertia helpers
Generate known-good shell cases (for example m=3, r=2 → I=8) plus deliberate negative-radius rejections for coding kata expected values in physics engines.
Technical questions
What does the hollow sphere 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.