Hollow Cylinder Moment of Inertia Calculator
A 12 kg tube with radii of 0.2 m inside and 0.5 m outside has a moment of inertia of 1.74 kg·m² about its long axis. Enter the total mass and both radii to see how the wall thickness shapes the result. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Mass
- 12
- Inner radius
- 0.2
- Outer radius
- 0.5
- Moment of inertia
- 1.74
- Axis
- central longitudinal axis
This hollow cylinder moment of inertia calculator finds rotational inertia about the cylinder's central longitudinal axis. Enter the total mass, inner radius, and outer radius in one consistent unit system.
Choose the correct hollow-cylinder model
Use this calculator for a uniform hollow circular cylinder rotating about the straight axis that passes through the centres of its circular ends. Typical examples include a thick pipe, a cylindrical sleeve, a tube with substantial wall thickness, or an idealised flywheel rim whose material fills the region between two radii. The required mass is the total mass of the cylinder, not density and not mass per unit length.
The inner radius measures from the axis to the inside wall, while the outer radius measures from the axis to the outside surface. Both radii should describe the same physical cross-section. This model assumes the material is distributed uniformly throughout that annular volume.
It does not calculate inertia about an axis across the diameter, about a displaced parallel axis, or for a cylinder with uneven density. A solid cylinder can be represented by setting the inner radius to zero, although a dedicated solid-cylinder formula may communicate that case more clearly. A very thin cylindrical shell is approached when the two radii become nearly equal.
Negative radii are invalid geometry and produce an input error rather than a numerical answer.
Enter consistent measurements and interpret the result
Keep the measurements in a coherent system because this calculator performs the mechanics formula but does not convert units. For example, kilograms for mass and metres for both radii produce a moment of inertia in kilogram square metres. Grams and centimetres instead produce gram square centimetres.
Mixing kilograms with one radius in millimetres and another in metres gives a numerically valid but physically meaningless result, so convert first. The calculation squares each radius, adds those squared distances, multiplies by the mass, and then takes one half. This makes the result especially sensitive to material farther from the axis: increasing the outer radius can change inertia much more than the same numerical change near the centre.
The response also echoes all three inputs, identifies the central longitudinal axis, gives the symbolic formula, and provides a unit hint. Zero mass returns zero inertia, and zero inner radius reduces the expression to the familiar solid-cylinder result.
Apply the value in rotational mechanics
Moment of inertia links an object's geometry and mass distribution to its rotational behaviour. After calculating I here, you can use torque equals I times angular acceleration to estimate the ideal torque needed for a target acceleration. You can also use one half I omega squared for rotational kinetic energy, provided angular speed omega is expressed consistently, or angular momentum equals I omega for rotation about the same fixed axis.
Keep the axis assumption attached to the number: a moment of inertia is never just a property of the object without reference to an axis. If the actual axis is parallel to the central axis but displaced by distance d, apply the parallel-axis theorem separately by adding m d squared to this calculator's central-axis result. For composite machinery, calculate each component about a common axis and sum the values, applying axis shifts where required.
Real components may contain spokes, hubs, holes, tapers, or density changes that violate the uniform thick-wall assumption; use a more detailed mass-property model when those features matter. This deterministic calculation is useful for homework checks, preliminary sizing, simulation inputs, and repeatable engineering scripts, but it does not replace measured or CAD-derived inertia for safety-critical design.
What you can do with it
Size a tubular rotor
Estimate central-axis inertia before comparing motor torque and angular acceleration requirements.
Check a mechanics exercise
Verify a textbook calculation for a uniform thick-walled cylinder using the stated mass and radii.
Prepare a simulation parameter
Generate a deterministic inertia value for an ideal hollow cylindrical component in a rotational model.
FAQ
Which axis does this calculator use?
It calculates inertia about the central longitudinal axis running through the centres of the circular ends.
Which units should I enter?
Use one consistent system. Kilograms and metres give kg·m²; grams and centimetres give g·cm².
Can either radius be negative?
No. A negative inner or outer radius is rejected as invalid input. Zero is allowed.
Can I use it for a solid cylinder?
Yes. Set the inner radius to zero and the formula becomes one half of the mass times the outer radius squared, which is the familiar result for a solid cylinder.
Should I enter the density or the mass?
The total mass of the cylinder, not its density and not its mass per metre. The calculator assumes that mass is spread evenly through the wall between the two radii.
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-cylinder-hollow \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"mass":12,"inner_radius":0.2,"outer_radius":0.5}'const res = await fetch("https://api.kit.forhosting.com/mech/moment-of-inertia-cylinder-hollow", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"mass": 12,
"inner_radius": 0.2,
"outer_radius": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/moment-of-inertia-cylinder-hollow",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"mass": 12,
"inner_radius": 0.2,
"outer_radius": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/moment-of-inertia-cylinder-hollow", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"mass":12,"inner_radius":0.2,"outer_radius":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"mass":12,"inner_radius":0.2,"outer_radius":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/moment-of-inertia-cylinder-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
{
"mass": 12,
"inner_radius": 0.2,
"outer_radius": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.moment_of_inertia_cylinder_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
The calculator applies the closed-form expression I = 0.5 × m × (inner radius squared + outer radius squared), then reports the result with the submitted dimensions and a unit reminder. It is designed for uniform, thick-walled circular cylinders and rejects negative radii instead of silently treating them as valid geometry.
Enter consistent measurements and interpret the result
Enter mass, inner_radius, and outer_radius as finite, non-negative numbers. The returned moment_of_inertia is the primary answer. Each successful API request has the base price $0.002; invalid input follows the typed error path.
Technical questions
What formula is used?
It uses I = 0.5 × m × (inner radius squared + outer radius squared) for a uniform hollow cylinder.
Can I calculate a solid cylinder?
Yes. Set inner_radius to zero and the equation reduces to I = 0.5 × m × outer_radius squared.
What does an API request cost?
Each successful request costs $0.002. You can also run the calculation free in your browser.