ForHosting KIT · Developer Utilities

Solid Shaft Polar Moment of Inertia Calculator

This solid shaft polar moment of inertia calculator finds the polar second moment of area, J, for a solid circular cross-section from one dimension: its diameter.

● BetaFree · in your browser

Run — free How it works & API

Use it from WebAPIEmailTelegramApp soon

The result is the geometric property used in elastic torsion and angle-of-twist equations. Enter a positive diameter, choose its length unit, and receive J in that unit raised to the fourth power. The calculation uses the exact circular-section relationship J = pi times diameter to the fourth power divided by 32.

What the polar moment means for a solid shaft

The polar moment of inertia, more precisely called the polar second moment of area, describes how a cross-section's area is distributed around an axis perpendicular to the section. For a solid round shaft, that axis passes through the center of the circle and follows the shaft length. A larger value of J indicates greater geometric resistance to twisting, but J is not a material property and does not by itself determine whether a shaft is safe. Material stiffness and strength enter through the shear modulus and allowable shear stress. This calculator applies only to a completely solid circular cross-section with one uniform outside diameter. It does not represent a tube, a keyway, a spline, a drilled bore, or a stepped region. Those shapes remove or redistribute material and need their own section calculation. The returned value is therefore best treated as a cross-sectional input for a shaft segment whose geometry genuinely matches a solid circle.

How to calculate J and keep units consistent

For diameter d, the calculator evaluates J = pi d^4 / 32. The fourth power is important: doubling the diameter multiplies J by sixteen, so even a small diameter change can strongly affect torsional behavior. Choose the unit that matches the entered diameter. If d is in millimetres, J is in mm^4; if d is in inches, J is in in^4. The tool labels that fourth-power unit but does not convert the diameter behind the scenes. When you place J into another engineering equation, every related quantity must belong to one consistent unit system. For example, using torque in newton-millimetres, length in millimetres, shear modulus in newtons per square millimetre, and J in mm^4 produces an angle of twist in radians. Do not mix a diameter entered in millimetres with a downstream formula expecting metres unless you first convert J by the correct fourth-power factor.

Using the result in torsion and twist checks

In elementary elastic torsion, the maximum shear stress at the outside surface of a solid circular shaft is tau_max = T c / J, where T is torque and c = d/2 is the outer radius. The angle of twist for a uniform segment is theta = T L / (G J), where L is segment length and G is the material's shear modulus. This calculator supplies J, one required term in both relationships. Before relying on either equation, confirm that the shaft is straight, circular, prismatic over the segment, and operating within the linear-elastic range. Account separately for stress concentrations near shoulders, grooves, holes, or keyways, and check applicable design codes, fatigue loading, deflection limits, and safety factors. For a shaft with multiple diameters, calculate each segment's J and sum each segment's contribution T L/(G J) when finding total twist. Engineering judgment remains necessary because a mathematically correct section property cannot capture omitted geometry, loading, or failure modes.

What you can do with it

Size a solid transmission shaft

Calculate J for candidate diameters before comparing elastic torsional stress and twist under a specified torque.

Check angle of twist

Use the returned section property with shaft length, torque, and shear modulus in the uniform-shaft twist equation.

Compare diameter alternatives

Quantify how strongly a diameter change affects torsional rigidity through the fourth-power relationship.

FAQ

What formula does the calculator use?

It uses J = pi d^4 / 32 for a solid circular cross-section, where d is the shaft diameter.

Is polar moment of inertia a mass property?

No. Here J is a geometric second moment of area with dimensions of length to the fourth power, not a mass moment of inertia.

Can I use this result for a hollow shaft?

No. A hollow circular shaft requires both outside and inside diameters and uses J = pi(D^4 - d^4)/32.

Why does the output unit have a fourth power?

The formula raises diameter to the fourth power, so millimetres produce mm^4 and inches produce in^4.

What does the API calculation cost?

Each API request costs $0.002. The browser calculator is available free on this page.

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

POSThttps://api.kit.forhosting.com/eng/polar-moi-circle

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/polar-moi-circle \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"diameter":50}'

const res = await fetch("https://api.kit.forhosting.com/eng/polar-moi-circle", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "diameter": 50
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/eng/polar-moi-circle",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "diameter": 50
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/eng/polar-moi-circle", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"diameter":50}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"diameter":50}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/eng/polar-moi-circle", 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": 50
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "eng.polar_moi_circle",
  "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

Per request$0.002

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.

Read the full KIT documentation →