Ellipsoid semi-axis from volume calculator
Recover the missing semi-axis of an ellipsoid from its volume — a volume of 8π with semi-axes 3 and 2 leaves a third semi-axis of 1. Useful when a tank note or textbook gives the capacity first. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Volume
- 25.1327412287
- A
- 3
- B
- 2
- C
- 1
A triaxial ellipsoid is fixed by three positive semi-axes a, b, and c, and its solid volume is exactly four-thirds pi times the product of those three lengths. Many textbooks, tank charts, and design notes give the filled volume together with only two of the three half-widths, then ask for the missing axis. Inverting the classical formula is a short algebraic step: multiply the volume by three, divide by four pi times the product of the two known semi-axes, and obtain the third. Students meet this reverse step when a problem states an ellipsoidal solid holds a given number of cubic units with two semi-axes listed and asks for the third; engineers and makers face the same algebra when a capacity note lists volume and two principal half-lengths.
What relating volume to a third semi-axis means for an ellipsoid
In Euclidean solid geometry the volume of a triaxial ellipsoid with semi-axes a, b, and c is four-thirds pi times the product a times b times c. When any one semi-axis is unknown and the volume plus the other two lengths are known, solve for the missing axis by rearranging: the third semi-axis equals three times the volume divided by four pi times the product of the two known semi-axes.
That map is unique and strictly positive whenever volume and both known axes are strictly positive, because the product of positive lengths is positive and the constant four-thirds pi is positive. Zero or negative volume has no meaning for a solid ellipsoid body, and a zero or negative known semi-axis would collapse the solid or reverse an orientation that the classical formula does not admit, so those inputs must be refused before any division is taken.
Search intent clusters around phrases like ellipsoid semi-axis from volume, find third axis of ellipsoid given volume, reverse ellipsoid volume calculator, missing semi-axis of triaxial ellipsoid, and c equals three V over four pi a b. People want a numeric third half-length they can check against a textbook key, a tank or pressure-vessel note that lists capacity first, or a CAD dimension list, not only a reminder that the solid is determined by three positive lengths.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate third semi-axis from a stated ellipsoid volume and two known half-lengths while checking homework, sizing an ellipsoidal tank or hull section from a capacity figure, verifying a CAD dimension from a catalog volume, or confirming a workshop note without installing a computer-algebra system. Results are fully deterministic: the same volume, a, and b always yield the same c.
There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Pair with a forward ellipsoid-volume tool when you also need surface area from three known axes; this capacity focuses on the reverse third-axis recovery path only.
What you can do with it
Homework drills that state volume and two axes
Students and graders recover the third semi-axis from a given ellipsoid volume and two known half-lengths, including clear rejection when any value is not positive.
FAQ
What happens if volume or a semi-axis is not positive?
Volume and both known semi-axes must be strictly greater than zero; those calls are not charged.
Is the ellipsoid semi-axis from volume calculator free?
The browser tool runs free on your device.
What units should I use for volume and semi-axes?
Use any consistent length unit for a and b; volume must be in the cube of that unit. The recovered c is returned in the same linear unit. The engine does not convert units.
What formula gives the missing semi-axis?
Multiply the volume by three and divide by four π times the product of the two known semi-axes. It is the classic ellipsoid volume, four thirds of π times a times b times c, solved for the third length.
Does it matter which known semi-axis I enter first?
No. Only the product of the two known semi-axes enters the calculation, so swapping them gives the same third semi-axis. What matters is that the volume is in the cube of the unit used for the lengths.
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/geom/ellipsoid-axis-volume \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"volume":25.132741228718,"a":3,"b":2}'const res = await fetch("https://api.kit.forhosting.com/geom/ellipsoid-axis-volume", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"volume": 25.132741228718,
"a": 3,
"b": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/ellipsoid-axis-volume",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"volume": 25.132741228718,
"a": 3,
"b": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/ellipsoid-axis-volume", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"volume":25.132741228718,"a":3,"b":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"volume":25.132741228718,"a":3,"b":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/ellipsoid-axis-volume", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"volume": 25.132741228718,
"a": 3,
"b": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.ellipsoid_axis_volume",
"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
This capacity accepts a strictly positive finite volume and two strictly positive finite semi-axes, rejects any non-positive value with a clear invalid_input error, and returns volume, a, b, recovered c, formula, and method as structured JSON.
What relating volume to a third semi-axis means for an ellipsoid
Delivering volume, a, b, c, formula, and method as JSON makes the result easy to paste into notes, diff in automated tests, or feed into a larger pipeline without reimplementing pi constants and inverse volume algebra in every language you teach or deploy.
How this calculator validates input and applies the inverse formula
On valid input the engine computes c as three times volume divided by four times pi times a times b. Extremely large volumes or axes above internal safety ceilings are refused so floating results stay finite and comparable. You may use any consistent unit system: if a and b are in meters then volume is in cubic meters and c is returned in meters, without unit conversion inside the engine. Send a JSON body with three required fields: volume, a, and b. Each value must be a finite number (JSON number or numeric string). The engine first checks that each field is present and finite; missing, empty, or non-numeric values fail with invalid_input naming the field. If volume is not strictly positive, or if either known semi-axis a or b is not strictly positive, the engine rejects the request with a clear message that volume and both known semi-axes must be positive, matching the product contract and common homework validation rules. Results are rounded to a stable decimal precision so the same input always yields the same JSON across JavaScript engines. Optional aliases semi_a or A for a and semi_b or B for b are accepted for convenience; this endpoint does not accept the third axis as an input because its job is the inverse recovery path only.
Practical use, limits, pricing, and delivery channels
Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted measures must be strictly positive finite numbers; a non-positive volume or axis is always rejected with invalid_input, and missing or non-finite measures are likewise refused. Wire the endpoint into solid-geometry drills that state volume first, contest answer checkers, ellipsoidal tank worksheets that list cubic capacity before the third half-width, or floating preprocessing steps where you already prefer explicit domain errors over opaque black-box widgets that cannot show why a non-positive value is rejected.
Integration use cases
Tank and vessel capacity notes to missing axis
Shop or logistics notes that list filled cubic capacity plus two principal half-lengths recover the matching third semi-axis as structured JSON for reports or drawings.
Deterministic fixtures for inverse solid-geometry tests
Pipelines that seed unit tests or interactive demos can call a pure third-axis-from-volume oracle instead of reimplementing the inverse (4/3)πabc algebra in every language.
Technical questions
How do I get the third ellipsoid semi-axis from volume with this API?
Send a JSON body with volume, a, and b (all strictly positive finite) to geom.ellipsoid_axis_volume. The response returns volume, a, b, recovered c, formula, and method.
Which formula does the calculator use?
From V = (4/3)·π·a·b·c it uses c = 3V/(4·π·a·b), summarized as c=3V/(4πab) from V=(4/3)πabc.