Octahedron circumradius calculator
An octahedron with edge 2 has a circumradius of exactly √2, the radius of the sphere touching all six vertices. Enter the edge to size a bounding sphere or check homework without mixing it up with the inradius. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Edge
- 2
- Circumradius
- 1.4142135624
- Circumsphere radius
- 1.4142135624
The circumradius of a regular octahedron is the radius of the circumsphere—the unique sphere that passes through every vertex of the solid. Students, CAD users, game artists, and exam writers often need that single length when sizing meshes, dimensioning crystal models, or comparing the classical inscribed and circumscribed spheres of a Platonic solid.
What the circumradius and circumsphere mean for a regular octahedron
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. It 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. Two concentric spheres are classically associated with that solid for homework and CAD sizing.
The insphere is tangent to the eight faces and has inradius proportional to square root of six over six times the edge. The circumsphere passes through all six vertices; its radius is the circumradius R. For edge length a the closed formula is R equals (a times square root of two) divided by two, which is algebraically identical to a divided by square root of two after rationalising.
That value is independent of orientation: only the common edge length matters. When a is zero the solid collapses and the circumradius is zero; when a is positive the circumsphere grows linearly with a. Negative a has no interpretation as a length, so this capability refuses it rather than taking an absolute value that would hide a data-entry bug.
Where the circumsphere radius is useful
Classrooms and tutoring platforms use an octahedron circumradius calculator so learners can check the circumscribed sphere radius for arbitrary non-negative edge lengths without mixing up inradius and midradius formulas. Exam-prep generators need a deterministic oracle that marks negative edges as invalid input instead of silently reflecting the absolute value and awarding partial credit by mistake. Failed validation does not consume balance.
Compared with a general regular-octahedron volume and multi-metric tool, this landing targets the narrower search intent of circumradius and circumsphere alone, so SEO pages do not compete only for volume or surface-area queries while still serving the exact scalar designers and students ask for when they type octahedron circumradius into a search box.
What you can do with it
Classroom checks of circumsphere radius
Let students submit an edge length and verify the circumradius R = a√2/2 against textbook worked examples without confusing inradius or volume formulas.
CAD, game assets, and bounding spheres
Compute the vertex sphere radius for regular octahedral cells in scripts that size collision shells, packing spheres, or crystal model dimensions.
Exam and LMS auto-grading
Mark circumradius items with a deterministic oracle that rejects negative edges and returns a stable twelve-decimal answer for non-negative edges.
FAQ
What is the difference between circumradius and circumsphere radius?
They are the same geometric length for a regular octahedron: the radius of the sphere that passes through all six vertices.
What happens if the edge is negative?
A negative edge length is not a valid geometric input and is not billed.
Does this return volume or inradius as well?
No. This capability focuses on circumradius only. For volume, surface area, and insphere radius from edge length, use the full regular octahedron calculator (geom.octahedron).
Is the edge times √2 over 2 the same as the edge divided by √2?
Yes. Rationalising the denominator turns one into the other, so both give the same circumradius. The radius also grows in step with the edge, so doubling the edge doubles the circumsphere too.
How do I use the circumradius to size a bounding sphere?
The circumsphere passes through all six vertices, so a sphere of that radius around the centre just contains the whole octahedron. That is the size to use for a collision shell, a packing sphere or a crystal model.
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-circumradius \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/octahedron-circumradius", {
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-circumradius",
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-circumradius", 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-circumradius", 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_circumradius",
"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 octahedron circumradius calculator accepts one edge length, applies the closed formula R equals a times square root of two over two, and returns both circumradius and circumsphere_radius as the same deterministic number. Negative edges are rejected with invalid_input so automation never treats a sign error as a physical solid. The free browser path and the paid API path share the same pure module, so classroom checks and production pipelines always agree on twelve-decimal rounding. At $0.002 per successful API call there is no model spend and no outbound network—only analytic arithmetic on one finite scalar.
What the circumradius and circumsphere mean for a regular octahedron
Reporting both circumradius and circumsphere_radius as the same field pair matches textbook language (circumradius) and CAD or mesh language (circumsphere radius) without forcing the client to rename keys.
Input contract, response shape, and error behaviour
Example: edge equals two yields circumradius equal to square root of two, approximately one point four one four two, because (two times square root of two) over two simplifies to square root of two. Edge equals one yields (square root of two) over two. Keep requests small, pin the rounded circumradius in automated tests, and you get reproducible vertex-sphere radii for homework, LMS plugins, and edge workers without standing up a full solid-geometry library of your own. POST a JSON body with one required field edge: a finite number or numeric string for the common edge length of the regular octahedron. Aliases a, side, edge_length, and length are accepted and mapped to the same parameter so notebook cells and form widgets can keep their local naming. Missing, empty, or non-finite values fail with invalid_input naming the field. On success the body echoes edge rounded to twelve decimal places, circumradius, circumsphere_radius (identical to circumradius), method set to regular_octahedron_circumradius, and formula string R = (a · √2) / 2 for display next to the answer. Pair the response with your own renderer if you need a 3D sketch of the circumsphere around the octahedron; this endpoint returns scalars only and does not emit meshes, unit conversions, volume, surface area, or inradius.
Where an octahedron circumradius API earns its keep
CAD, digital-fabrication, and game-asset notebooks often size bounding spheres or collision shells around regular octahedral cells; calling a tiny parse API avoids pulling a computer-algebra system for one multiply by square root of two over two.
Technical questions
How do I calculate the circumradius of a regular octahedron?
Send a finite non-negative edge length as edge (or a, side, edge_length, length). The API returns circumradius = (edge · √2) / 2 and the same value as circumsphere_radius.
What does the octahedron circumradius calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.