Stellated octahedron calculator
Enter one edge and get the volume and outer surface of the stella octangula, Kepler's star of two interlocking tetrahedra: an edge of 1 gives a volume of 0.176776695297 and a surface of 2.598076211353. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Edge
- 1
- Volume
- 0.1767766953
- Surface area
- 2.5980762114
A stellated octahedron, also called the stella octangula, is the compound of two dual regular tetrahedra and the unique stellation of the regular octahedron. Johannes Kepler named it in the early seventeenth century, yet the same star solid appears in modern solid-geometry homework, sacred-geometry sketches, game-asset notes, and crystal or jewelry models whenever someone records only a single outer edge length. Once that positive edge is fixed, closed-form formulas give the volume of the solid union and the exterior surface area of the eight triangular spikes without iterative mesh sampling. Volume equals the square root of two over eight times the cube of the edge; surface area equals three-halves times the square root of three times the square of the edge.
What a stellated octahedron is and which metrics the edge determines
In Euclidean solid geometry the stellated octahedron is completely determined, up to congruence, by a single positive edge length: the common edge of the two dual regular tetrahedra that form the compound, also equal to the outer ridge length of each star point. The figure can be built by attaching a regular tetrahedral spike to every face of a regular octahedron whose own edge is half that outer length, or by interpenetrating two dual tetrahedra so their intersection is that central octahedron.
Volume of the solid union equals the square root of two divided by eight, times the edge cubed. Exterior surface area sums twenty-four small equilateral triangles of side half the outer edge—three exposed faces on each of eight spikes—which simplifies to three-halves root-three times the edge squared. Search intent clusters around phrases like stellated octahedron calculator, stella octangula volume, stellated octahedron surface area, and star tetrahedron edge metrics.
People want numeric results they can check against a textbook answer key, a CAD note, or a game-asset size chart, not only a geometric argument that the star is determined. Related tools such as a plain regular-octahedron calculator answer a different convex-solid query; this page is dedicated to the stellated form alone.
How this calculator validates input and applies the closed-form formulas
The value must be a positive finite number, optionally written as a numeric string that parses cleanly. When the edge passes those checks, volume is computed as the square root of two over eight times the edge cubed, and surface area as three times the square root of three over two times the edge squared. Extremely small edges below the internal stability floor and extremely large edges above a hard upper bound are rejected rather than returning non-finite floats.
You may use any consistent unit system for the edge; volume and area are expressed in the corresponding cubic and square units without conversion inside the engine, which keeps homework and CAD workflows predictable. If you only know the edge of the inner regular octahedron, double it to obtain the outer tetra edge before calling this capacity, because the published formulas here are parameterized by that outer stella edge.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate stellated-octahedron solution while checking homework, sizing a star-tetrahedron model, verifying a game mesh scale from an edge length, or confirming a workshop drawing without installing a computer-algebra system. Results are fully deterministic: the same edge always yields the same volume and surface area. 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.
The capacity does not convert units, draw diagrams, or solve irregular compounds; those remain separate tools or manual steps outside this contract.
What you can do with it
Homework and exam stella octangula drills
Students and graders confirm volume and surface area of a stellated octahedron from a single positive edge, including rejection when the value is negative or otherwise invalid.
FAQ
Which formulas does the calculator use?
Volume is (√2/8)·edge³ for the solid union of the two dual tetrahedra. Surface area is (3√3/2)·edge² from the eight spikes of three equilateral triangles of side edge/2 each. Both assume a regular stella octangula.
What happens if the edge is negative?
A stellated octahedron requires a strictly positive edge; negative values are rejected and those calls are not charged.
Is the stellated octahedron calculator free?
The browser tool runs free on your device.
How is this different from a regular octahedron calculator?
A regular octahedron is the convex Platonic solid with eight faces. The stellated octahedron is the star compound (stella octangula) built from two dual tetrahedra or from octahedron face spikes; its volume and surface formulas differ and answer a different search intent.
Which edge of the stella octangula should I measure?
Use the common edge of the two tetrahedra, which is also the outer ridge of each star point. The octahedron hidden at the centre has an edge exactly half that length.
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/stellated-octahedron \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/stellated-octahedron", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"edge": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/stellated-octahedron",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"edge": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/stellated-octahedron", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"edge":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"edge":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/stellated-octahedron", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"edge": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.stellated_octahedron",
"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 edge, rejects negative and non-finite values with a clear invalid_input error, and returns volume and surface area as stable structured JSON.
What a stellated octahedron is and which metrics the edge determines
Delivering edge, volume, and surface_area as JSON makes the result easy to paste into notes, diff in automated tests, or feed into a larger pipeline without reimplementing roots and multiplications in every language you teach or deploy.
How this calculator validates input and applies the closed-form formulas
Send a JSON body with a single required field named edge. The engine rejects missing input, empty strings, non-numeric values, zero, and especially negative numbers before any metric is computed, and reports invalid_input so failed validation is not charged. Results are rounded to a stable decimal precision so the same input always yields the same JSON across JavaScript engines.
Practical use, limits, pricing, and delivery channels
Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted inputs must be strictly positive; a negative edge is rejected with a clear invalid_input message that states the value cannot be negative. Wire the endpoint into solid-geometry drills, contest answer checkers, Kepler-solid worksheets that list only the edge, or floating preprocessing steps where you already prefer explicit domain errors over opaque black-box widgets that cannot show why a negative measure is rejected.
Integration use cases
Deterministic fixtures for star-polyhedron tests
Pipelines that seed unit tests or interactive Kepler-solid demos can call a pure stellated-octahedron oracle instead of reimplementing roots in every language.
Game assets and crystal models from edge length
Artists and engineers who know only the outer edge of a star tetrahedron recover volume and surface area as structured JSON for reports, mesh scales, or lightweight CAD helpers.
Technical questions
How do I compute stellated octahedron metrics with this API?
Send a JSON body with a positive edge field to geom.stellated_octahedron. The response returns edge, volume, surface_area, method, and formula strings.