Polar moment of inertia calculator
A solid shaft 0.1 m across has a polar moment of about 9.817 × 10⁻⁶ m⁴. Enter the outer diameter in metres to get J = πd⁴/32, the figure that goes into torsional shear stress and angle of twist. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Diameter
- 0.1
- Polar moment
- 0.0000098175 m^4
The polar moment of inertia of a solid circular shaft is the second moment of area about the longitudinal axis, and it is the geometric J that appears in every elastic torsion formula relating torque to shear stress or angle of twist. For a circular cross-section it collapses to a closed form in the outer diameter alone: J equals pi times d to the fourth over thirty-two, reported in metres to the fourth power when d is in metres.
What the polar moment says about a shaft
The polar moment of inertia, often written J or I_p, is a purely geometric property of a cross-section: it is the second moment of area about an axis perpendicular to the plane of the section through its centroid, and by the perpendicular-axis theorem it equals I_x plus I_y. For a circular disc that identity collapses because I_x equals I_y equals π d⁴ / 64, so J equals twice I_x equals π d⁴ / 32. That number is not a mass moment of inertia.
Mass moment of inertia has units of kilogram square metres and describes how a rigid body resists angular acceleration; polar second moment of area has units of metres to the fourth power and describes how a cross-section resists torsion. Searchers who want ring, disc, sphere, or rod mass inertia are on a different landing; this page only answers the circular-shaft polar J from diameter. In strength of materials, J is the denominator of the elastic torsion formulae.
The shear stress at radius r is τ = T r / J, so the maximum shear on a solid round bar is at the outer fibre: τ_max = T (d/2) / J, which becomes 16 T / (π d³) once J is substituted. The angle of twist over length L is θ = T L / (G J), where G is the shear modulus.
Both identities fail if J is computed from the wrong diameter, from millimetres treated as metres, or from a hollow-tube formula applied to a solid bar. This calculator implements only the solid circular case: one diameter, J = π d⁴ / 32. It does not subtract an inner bore, does not convert millimetres, and does not solve for torque or twist.
A very small positive diameter is valid and yields a correspondingly small J; it is not treated as zero. If you have a hollow shaft, form J = π (D⁴ − d⁴) / 32 yourself from outer and inner diameters—this page does not accept a bore. Rectangular, elliptical, and open sections need other closed forms or a numerical polar moment; they are outside this contract.
How diameter is read, which units apply, and why zero fails
Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; embedded spaces, hex prefixes, and non-numeric tokens are rejected with a clear message so a bad cell does not become a wrong polar moment. Negative diameters are likewise rejected; a signed width has no meaning for a solid circular bar. An infinite, NaN, or otherwise non-finite diameter is rejected, as is a magnitude above the published ceiling, so a runaway millimetre-to-metre mix-up cannot overflow into a non-finite J.
Units are SI and are not converted. The polar moment is always returned in metres to the fourth power. Computation uses the host Math.PI constant times d to the fourth divided by thirty-two, then rounds to twelve significant digits and collapses negative zero to plain zero for clean comparisons.
Keep requests inside the documented absolute range. A diameter of one metre is a convenient check case: J equals π / 32, about 0.098175 m^4. If your drawing lists inches, convert to metres first by multiplying by 0.0254.
Where circular-shaft polar moment calculations earn their keep
Machine-design students meet polar moment of inertia the first time a textbook asks for the shear stress in a driven shaft: they are given torque and diameter and must form J before they can write τ = T r / J. Lecturers can pre-generate answer keys for modest diameters without trusting opaque calculator apps that silently switch millimetres and metres.
Developers call the same contract from scripts to seed deterministic test vectors for torsion demos, to normalize user-entered shaft specs before UI plots of twist or shear, or to satisfy programming-challenge statements that demand J rather than a hand-waved sketch. Contest practice often needs a fast, offline source of solid-circle J values so solutions can be checked without graphing calculators under time pressure. Product and education tools use the closed form when a learner types a diameter and the UI must show J and a clean formula reminder.
Pair it with a mass-moment calculator when the search is I = k m r² for a rigid body, and with a section-modulus converter when you already have J and only need another unit of length to the fourth. Do not use this page for hollow tubes, keyed shafts, or non-circular sections: those geometries change the closed form and would silently mis-size a shaft if forced through π d⁴ / 32.
Maintenance and reliability scripts can batch diameters from a bill of materials, reject rows with missing or non-positive d, and attach J to each solid round bar before a torsion check.
What you can do with it
Shaft torsion and machine design
Compute J = π d⁴ / 32 for a solid round shaft so elastic torsion checks τ = T r / J and twist θ = T L / (G J) use the correct geometric polar moment.
FAQ
How is the polar moment of a circular shaft computed?
Send a finite positive diameter d in metres.
What happens if the diameter is zero or negative?
Polar moment of inertia J = π d⁴ / 32 is undefined for a zero or negative shaft diameter, and those failures are not billed.
What units does the polar moment calculator use?
Diameter must be in metres. The polar moment is always returned in m^4. Convert millimetres (divide by 1000) or inches (multiply by 0.0254) before calling; the engine does not convert units.
Is this the same as mass moment of inertia?
No. Polar moment of inertia here is the second moment of area (m^4) used in torsion of shafts. Mass moment of inertia (kg·m²) for rings, discs, spheres, and rods is a different search and a different calculator.
Why does a slightly thicker shaft resist twisting so much more?
Because J grows with the fourth power of the diameter. Doubling the diameter multiplies J by 16, so a modest increase in size gives a much stiffer, stronger shaft in torsion.
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/phys/polar-moment \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"diameter":0.1}'const res = await fetch("https://api.kit.forhosting.com/phys/polar-moment", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"diameter": 0.1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/polar-moment",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"diameter": 0.1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/polar-moment", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"diameter":0.1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"diameter":0.1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/polar-moment", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"diameter": 0.1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.polar_moment",
"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_diameter | 1000000 |
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
This calculator accepts a single positive finite diameter, applies that identity with deterministic floating-point arithmetic, and returns the polar moment together with the formula string so a worksheet, a shaft-sizing script, or a lecture demo can reuse the same number the API would charge for. A zero or negative diameter is rejected with invalid_input, because a shaft that has no positive width cannot have a well-defined polar second moment. Successful API calls are billed at $0.002; invalid inputs are not charged.
What J = π d⁴ / 32 means and what the response contains
Send the outer diameter in metres; read polar_moment in m^4. The response echoes the normalized diameter, the polar_moment, a method tag solid_circular, the formula string J = π · d^4 / 32, and the unit m^4. There is no Monte Carlo sampling of the section, no CAD kernel, and no random rounding: the engine either evaluates π d⁴ / 32 inside the supported magnitude range or rejects the input with invalid_input. Rejected cases include a missing diameter, a non-finite value, a magnitude above the documented ceiling, and a diameter that is zero or negative.
How diameter is read, which units apply, and why zero fails
The preferred field is diameter for the outer diameter in metres, but aliases d and D are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. Diameter must already be in metres; millimetres must be divided by one thousand before the call (a twenty millimetre shaft is 0.02 m, a fifty millimetre shaft is 0.05 m, a one hundred millimetre shaft is 0.1 m). Mixing millimetres into d without converting inflates J by a factor of ten to the twelfth, which is a classic worksheet error this API will not silently correct. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline. A one hundred millimetre shaft (d = 0.1 m) yields π times 10^−4 over 32, about 9.817e-6 m^4. If you need both J and the area moment I_x for a round bar, remember I_x = J / 2 = π d⁴ / 64; this endpoint returns only J.
Where circular-shaft polar moment calculations earn their keep
On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Keep the diameter in metres, pin the rounded polar_moment in tests, and you get reproducible J for homework, LMS plugins, and edge workers without standing up a finite-element mesh for a circular bar that already has a closed form.
Integration use cases
Homework and lecture keys
Give students a deterministic polar-moment oracle from diameter in metres, including a hard invalid_input path when d is zero or negative.
Deterministic test fixtures
Generate reproducible polar_moment values in m^4 for unit tests, property checks, and CI assertions on torsion or shaft-sizing code.
Technical questions
What does the polar 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.