Octahedron calculator
One edge length fixes every measure of a regular octahedron: with an edge of 2 the circumradius is 1.4142135624 and the volume 3.7712361663. Get volume, surface and both sphere radii together for homework, crystal models or game assets. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Edge
- 2
- Volume
- 3.7712361663
- Surface area
- 13.8564064606
- Inradius
- 0.8164965809
- Circumradius
- 1.4142135624
A regular octahedron is one of the five Platonic solids: eight congruent equilateral triangular faces, twelve equal edges, and six vertices where four faces meet. Once a single positive edge length is fixed, every classical metric of the solid is determined by closed-form formulas. Volume is the square root of two over three times the cube of the edge, total surface area is twice the square root of three times the square of the edge, the inradius of the inscribed sphere is the edge times the square root of six over six, and the circumradius of the sphere through all vertices is the edge times the square root of two over two. Students meet these formulas in solid-geometry homework and standardized tests; makers, game artists, and engineers need the same numbers when a sketch or asset note lists only the edge.
What a regular octahedron is and which metrics the edge determines
In Euclidean solid geometry a regular octahedron is completely determined, up to congruence, by a single positive edge length. All twelve edges share that length. The solid can be viewed as two square pyramids glued base-to-base, or as the dual of the cube, with vertices along the coordinate axes when centered at the origin.
Volume equals the square root of two divided by three, times the edge cubed. Surface area sums eight equilateral triangles, each of area one-fourth root-three times the edge squared, which simplifies to twice root-three times the edge squared. The insphere touches every face at its centroid; its radius is the edge times root-six over six.
The circumsphere passes through all six vertices; its radius is the edge times root-two over two. Search intent clusters around phrases like octahedron calculator, regular octahedron volume, octahedron surface area, octahedron inradius, and octahedron circumradius from edge. 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 figure is determined.
Related solids such as cubes or spheres answer different queries; this page is dedicated to the regular octahedron edge problem 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 three times the edge cubed, surface area as twice the square root of three times the edge squared, inradius as the edge times the square root of six over six, and circumradius as the edge times the square root of two over two.
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, area, and radii are expressed in the corresponding cubic, square, and linear units without conversion inside the engine, which keeps homework and CAD workflows predictable.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate regular-octahedron solution while checking homework, sizing a gem or crystal 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, surface area, inradius, and circumradius. 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 octahedra; those remain separate tools or manual steps outside this contract.
What you can do with it
Homework and exam octahedron drills
Students and graders confirm volume, surface area, inradius, and circumradius from a single positive edge, including rejection when the value is not positive.
FAQ
Which formulas does the calculator use?
Volume is (√2/3)·edge³. Surface area is 2·√3·edge². Inradius is edge·√6/6. Circumradius is edge·√2/2. All formulas assume a regular octahedron with equal edges and equilateral faces.
What happens if the edge is not positive?
A regular octahedron requires a strictly positive edge; zero and negative values are rejected and those calls are not charged.
Is the octahedron calculator free?
The browser tool runs free on your device.
What is the difference between inradius and circumradius?
The inradius is the radius of the inscribed sphere that touches every face. The circumradius is the radius of the circumscribed sphere that passes through all six vertices; for a positive edge it is always larger than the inradius.
How is a regular octahedron related to a cube?
It is the dual of the cube: centred at the origin, its six vertices lie on the coordinate axes. You can also picture it as two square pyramids joined at their bases.
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/octahedron \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/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/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/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/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.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 non-positive or non-finite values, and returns the full set of metrics as structured JSON.
What a regular octahedron is and which metrics the edge determines
Delivering edge, volume, surface_area, inradius, and circumradius 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 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; zero-edge or negative-edge octahedra are rejected with a clear invalid_input message. Wire the endpoint into solid-geometry drills, contest answer checkers, Platonic-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 non-positive measure is rejected.
Integration use cases
Deterministic fixtures for solid-geometry tests
Pipelines that seed unit tests or interactive Platonic-solid demos can call a pure octahedron oracle instead of reimplementing roots in every language.
Game assets and crystal models from edge length
Artists and engineers who know only the edge of a regular octahedron recover full metrics as structured JSON for reports, mesh scales, or lightweight CAD helpers.
Technical questions
How do I compute regular octahedron metrics with this API?
Send a JSON body with a positive edge field to geom.octahedron. The response returns edge, volume, surface_area, inradius, and circumradius.