Ellipsoid volume calculator
Enter three semi-axes and get the ellipsoid's volume and skin area together: semi-axes 3, 2 and 1 give a volume of 8π. The area uses the Knud Thomsen approximation, exact for a sphere. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 3
- B
- 2
- C
- 1
- Volume
- 25.1327412287
- Surface area
- 48.9719333337
- Surface area method
- knud_thomsen
- Thomsen p
- 1.6075
A triaxial ellipsoid is the solid you get by stretching a unit sphere independently along three mutually perpendicular axes. It models planets, eggs, rugby balls, tank ends, and many CAD primitives more faithfully than a pure sphere when the three principal lengths differ. Given positive semi-axes a, b, and c measured in the same length unit, the volume has a simple closed form: four thirds of π times a times b times c. Surface area has no elementary closed form for the general case; the practical engineering standard is the Knud Thomsen approximation, which raises pairwise products of the powered axes, averages them, takes the p-th root with p equal to 1.6075, and multiplies by four π. Relative error stays small for moderate eccentricity and the formula collapses to the exact sphere area when a equals b equals c.
What volume and Knud Thomsen surface area mean for an ellipsoid
A triaxial ellipsoid is defined by three positive semi-axes a, b, and c along orthogonal principal directions; every point on the surface satisfies x squared over a squared plus y squared over b squared plus z squared over c squared equals one. Volume follows at once from the linear stretching of a unit ball: V equals four thirds π a b c. When a equals b equals c equals r, this reduces to the familiar sphere volume four thirds π r cubed.
Surface area is harder: the exact expression involves incomplete elliptic integrals of the first and second kinds and does not simplify for arbitrary a, b, and c.
Knud Thomsen proposed a compact approximation that is widely used in engineering tables and software: S approximately equals four π times the quantity open parenthesis a to the p times b to the p plus a to the p times c to the p plus b to the p times c to the p, all over three, close parenthesis, raised to the power one over p, with the fixed exponent p equal to 1.6075.
Units stay consistent with whatever length unit you pass for a, b, and c—metres yield cubic metres of volume and square metres of area; inches yield cubic and square inches. The arithmetic is a handful of multiplications and one power after coercion, so there is no iteration, no Monte Carlo sampling, and no dependence on wall-clock time.
What you can do with it
Tank ends, eggs, and packaging takeoff
Turn principal semi-axes into volume for capacity estimates, or approximate skin area for paint, wrap, and material orders on ellipsoidal parts.
CAD and manufacturing pre-checks
Validate stock ellipsoid dimensions in scripts before nesting or quoting: volume for mass estimates, approximate surface for coating and heat treatment.
FAQ
What is the formula for the volume of an ellipsoid?
With semi-axes a, b, and c, volume is (4/3)·π·a·b·c. All three lengths must use the same unit. When a=b=c=r this is the sphere volume (4/3)·π·r³.
Is the ellipsoid volume calculator free?
The browser tool is free and runs locally.
How accurate is the ellipsoid surface area?
It is exact for a sphere and usually stays within about one percent for moderate shapes. Very elongated, needle-like or flattened, pancake-like ellipsoids can drift further, which is why the area is labelled as an approximation.
What units do the volume and area come out in?
In the unit you use for the three semi-axes, cubed for the volume and squared for the area. Metres give cubic and square metres, inches give cubic and square inches, and all three must share one unit.
What happens when all three semi-axes are equal?
The ellipsoid becomes a sphere of that radius. The volume reduces to four thirds of π times the radius cubed, and the Knud Thomsen area turns into the exact surface of that sphere.
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 \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":3,"b":2,"c":1}'const res = await fetch("https://api.kit.forhosting.com/geom/ellipsoid", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 3,
"b": 2,
"c": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/ellipsoid",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 3,
"b": 2,
"c": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/ellipsoid", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":3,"b":2,"c":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":3,"b":2,"c":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/ellipsoid", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 3,
"b": 2,
"c": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.ellipsoid",
"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_semi_axis | 1000000000000 |
min_semi_axis | 1e-15 |
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 capability packages both results as deterministic JSON with no computer-algebra system, no random seeds, and the same pure solve function in the free browser tool and the prepaid API path. Any non-positive semi-axis is rejected with invalid_input so scripts and grading pipelines see an honest error instead of a silent absolute value or a zero-volume solid.
What volume and Knud Thomsen surface area mean for an ellipsoid
The approximation is exact for a sphere and stays within roughly one percent relative error for many practical eccentricities; extreme needle-like or pancake-like shapes can deviate more, which is why the response labels the surface field as a method knud_thomsen rather than exact. Zero or negative axes are not accepted under this contract because they do not describe a solid ellipsoid of positive extent; the API returns invalid_input rather than inventing a magnitude.
How to call the API and what comes back
All three must be finite numbers strictly greater than zero and inside the published limits, and they must share the same length unit. Use it from geometry homework pipelines, planetary or tank takeoff scripts, interview-prep sandboxes, CAD pre-checks, or any workflow that needs reproducible ellipsoid numbers without installing a computer-algebra package on every machine. Send a JSON body with the three semi-axes a, b, and c. Field a is the first positive semi-axis; aliases semi_a, A, and x are accepted. Field b is the second; aliases semi_b, B, and y map to the same meaning. Field c is the third; aliases semi_c, C, and z are accepted. On success the response echoes a, b, and c; returns volume as the exact four thirds π a b c value; returns surface_area from the Knud Thomsen formula; sets surface_area_method to knud_thomsen and thomsen_p to 1.6075; flags is_sphere when the three axes are equal; sets method to triaxial_ellipsoid; and includes a short formula block documenting both expressions. Failed validation—missing a required field, a non-finite number, a non-positive axis, or a value outside published limits—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price. Pair the call with unit conversion only when your source data mixes millimetres and metres; this capability does not convert linear units for you, so keep a, b, and c in the same unit before you send the request.
Practical tips, limits, and related searches
When teaching, prefer paired examples: a three, b two, c one for a clearly triaxial solid; a one, b one, c one for a unit sphere where volume is four thirds π and surface area is four π; and a zero or negative axis to demonstrate the hard rejection path. Prefer this tool when you already know the three principal semi-axes and need volume plus a standard approximate surface in one shot. Round-trip checks are easy: after solving a sphere case, confirm volume equals four thirds π r cubed and surface equals four π r squared; for unequal axes, confirm volume scales with the product a b c when you double one axis. If you need only a right circular cylinder from radius and height, use the dedicated cylinder volume calculator; if you need a regular tetrahedron from edge length, use the tetrahedron tool instead of forcing degenerate ellipsoid axes. Keep each semi-axis inside the published positive range up to one trillion so intermediate products and powers stay finite in IEEE double precision on the Worker and in the browser. Because volume only needs the product a b c, students can recompute by hand for nice fractions of π and match the API line by line after dividing reported floats by π if desired. Surface area checks against a sphere are the best sanity test: set all three axes equal and confirm surface_area matches four π r squared within floating tolerance. This landing targets people searching for an ellipsoid volume calculator or Knud Thomsen surface area from three semi-axes; it is not a cylinder, tetrahedron, or exact elliptic-integral solver. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible measures while learning the contract.
Integration use cases
Homework checks for ellipsoid volume and surface area
Give students three semi-axes; compare their hand-computed volume and Knud Thomsen surface against a deterministic API or free browser result.
Technical questions
What is the Knud Thomsen surface area formula?
Approximate surface area is 4·π·((a^p·b^p + a^p·c^p + b^p·c^p)/3)^(1/p) with p=1.6075. It is exact for a sphere and a standard engineering approximation for triaxial ellipsoids.
Why are zero or negative semi-axes rejected?
A solid ellipsoid needs strictly positive semi-axes. Non-positive input returns invalid_input instead of taking an absolute value or returning a zero-volume solid.
Is the surface area exact?
Volume is exact. Surface area uses the Knud Thomsen approximation (p=1.6075); the true surface requires elliptic integrals and is not computed here.