Parallel-axis theorem inertia calculator
Shift a section's moment of inertia to a new parallel axis: 200 about the centroid, with an area of 50 and an offset of 4, becomes 1000, because the transfer term adds 800. It works with area, not mass. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- I centroid
- 200
- Area
- 50
- Distance
- 4
- Distance squared
- 16 m⁴
- Transfer term
- 800
- I parallel
- 1,000
The parallel-axis theorem, also called Steiner’s theorem, moves a section moment of inertia from a centroidal axis to any parallel axis once you know the cross-sectional area and the perpendicular distance between those axes. Strength-of-materials students, structural engineers, and machine designers need that transfer constantly when a handbook lists Ī about the centroid but the bending axis sits at a flange, a weld, or a built-up neutral axis. This parallel-axis theorem inertia calculator accepts the centroidal second moment Ī, the area A, and the offset d, then returns I equals Ī plus A times d squared in coherent length-to-the-fourth units.
What the parallel-axis theorem states and why I = Ī + A · d²
A plane section has a second moment of area, often written Ixx or Iyy, that measures how its area is distributed about an axis in its plane. That property is smallest about an axis through the centroid, conventionally written Ī or I_bar. If you need the same section’s inertia about a different axis that stays parallel to the first, the parallel-axis theorem adds a transfer term: I equals Ī plus A times d squared, where A is the cross-sectional area and d is the perpendicular distance between the two axes.
The extra A d² is exactly the contribution of treating the whole area as concentrated at the centroid and then moving that point mass of area a distance d from the new axis. The identity is used every time a rolled shape is welded off the neutral axis of a built-up beam, a hole is shifted from the centroid of a plate, or a rectangle is taken about its base instead of its mid-height.
It is not a unit conversion and it is not a shape-factor table: you must already know Ī about a centroidal axis of the piece you are moving. The mass-inertia cousin I equals Icm plus m d² uses mass instead of area and answers a different question (rotational kinetics). This calculator implements only the section form, so a beam or column worksheet can transfer Ixx or Iyy without mixing kilograms into millimetres-to-the-fourth.
Zero offset correctly leaves I equal to Ī; a vanishing area is rejected because there is then no section to transfer.
Where moving a section's inertia helps
Steel and timber designers move catalogue centroidal inertias onto the neutral axis of a built-up girder when adding cover plates or sistered members. Content sites that rank for parallel axis theorem calculator, section moment of inertia transfer, or Steiner theorem I equals I bar plus A d squared benefit from a live free tool whose arithmetic matches the paid automation path byte-for-byte.
Laboratory benches that measure second moments of cut sections can compare the transferred I against a physical pendulum or a bending test without embedding ad-hoc cells that disagree on rounding. Invalid input is not billed.
What you can do with it
Built-up beam and girder design
Move catalogue centroidal inertias onto a composite neutral axis when adding cover plates, sistered joists, or welded flange plates.
FAQ
Is this the mass form I = Icm + m d²?
No. This calculator transfers a section (area) second moment. The mass-inertia theorem uses mass m, not area A, and answers rotational kinetics rather than beam bending.
What units should I use?
Any coherent length system: metres give m⁴; millimetres give mm⁴. Keep Ī, A, and d in the same length unit. The response labels unit as m⁴ for the SI convention.
Why must the first axis be centroidal?
The theorem I=Ī+A·d² holds only when Ī is taken about an axis through the centroid of A. Transferring between two arbitrary parallel axes requires an extra step through the centroid.
How does the parallel axis theorem work for a beam section?
Add the area times the offset squared to the centroidal moment of inertia. With 200 about the centroid, an area of 50 and an offset of 4, you get 200 plus 50 times 16, which is 1000.
Can it work out the centroidal inertia from width and height?
No. Take that value from a handbook or your own shape formula first; this tool only moves an inertia you already know to a parallel axis. Keep every input in the same length system.
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/eng/parallel-axis \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"i_centroid":200,"area":50,"distance":4}'const res = await fetch("https://api.kit.forhosting.com/eng/parallel-axis", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"i_centroid": 200,
"area": 50,
"distance": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/eng/parallel-axis",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"i_centroid": 200,
"area": 50,
"distance": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/eng/parallel-axis", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"i_centroid":200,"area":50,"distance":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"i_centroid":200,"area":50,"distance":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/eng/parallel-axis", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"i_centroid": 200,
"area": 50,
"distance": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "eng.parallel_axis",
"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
Invalid input—missing fields, negative inertia, non-positive area, or a negative distance—fails with invalid_input and is not billed.
Input contract, units, response shape, and error behaviour
Numeric results are rounded to twelve decimal places for stable goldens and automation. Example: centroidal inertia two hundred, area fifty, distance four yields transfer term eight hundred and parallel-axis inertia one thousand, because two hundred plus fifty times sixteen equals one thousand. Pair the result with your own shape formulas if you still need Ī from width and height; this capability only transfers an already known section inertia. POST a JSON body with three required finite numbers: i_centroid (aliases I_bar, Ibar, i_bar, Icm, Ig, I0, ic, centroidal_inertia), the non-negative section moment of inertia about the centroidal axis; area (aliases A, section_area), a strictly positive cross-sectional area; and distance (aliases d, offset, axis_distance), a non-negative perpendicular offset between the two parallel axes. When Ī is in metres to the fourth, area in square metres, and d in metres, i_parallel is in m⁴; the same arithmetic holds for millimetres or inches if every input stays in one length system. Fields may arrive as JSON numbers or numeric strings. Missing fields, non-finite values, negative i_centroid, non-positive area, or a negative distance fail with invalid_input naming the problem; those failures are not billed. On success the body echoes rounded i_centroid, area, and distance; reports distance_squared and the transfer_term A times d squared; returns i_parallel as Ī plus that transfer term; sets unit to m⁴ as the SI default label; and includes formula I = Ī + A · d², method parallel_axis_theorem, and a short human-readable message.
Where a parallel-axis section inertia API earns its keep
Strength-of-materials courses and professional exams routinely ask for I about a base, a weld line, or a composite neutral axis after Ī and A are known; a deterministic oracle marks negative area or a signed offset as invalid_input instead of silently squaring a bad d. Finite-element pre-processors and spreadsheet replacements can pin i_parallel in continuous integration so a later mesh or hand calc cannot drift from Steiner’s identity. Keep requests small, pin the rounded i_parallel field in tests, and you get reproducible parallel-axis transfers for LMS plugins, edge workers, and design notebooks without standing up a full section-properties library.
Integration use cases
Homework and exam checks
Transfer Ī to a parallel axis with I=Ī+A·d² from textbook area and offset; reject negative area or signed d so graders get a clear invalid_input signal.
Lab and automation scripts
Pin i_parallel from known Ī, A, and d in CI using the same pure module as the free browser widget.
Technical questions
How do I apply the parallel-axis theorem to a section moment of inertia?
Send non-negative i_centroid Ī, positive area A, and non-negative distance d. The API returns I=Ī+A·d² as i_parallel in coherent length⁴ units (m⁴ when SI).
What does the parallel-axis theorem 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.