Icosahedron circumradius calculator
An icosahedron with an edge of 2 fits inside a sphere of radius about 1.9021 that touches all 12 vertices. Enter the edge and get that circumradius, for answer keys, crystal models and bounding spheres. 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
- 1.9021130326
A regular icosahedron is a Platonic solid with twenty congruent equilateral triangular faces, twelve vertices where five faces meet, and thirty equal edges. When only the common edge length is known, the radius of the sphere that passes through every vertex—the circumradius—is fixed by a closed radical form involving the square root of five: edge over four, times the square root of ten plus two times square root of five. Students checking solid-geometry homework, exam writers who need a deterministic answer key, makers sizing geodesic and crystal models from a measured edge, and game or CAD pipelines that store regular icosahedra by edge length all need that single number without re-deriving nested radicals by hand.
What the circumradius of a regular icosahedron means and how the formula works
In solid geometry the circumradius of a polyhedron is the radius of the unique sphere that passes through every vertex of the solid. For a regular icosahedron that sphere is uniquely determined once the common edge length is fixed. Elementary vector geometry shows that the distance from the center to any vertex equals a over four times the square root of ten plus two times square root of five, which is exactly the circumradius formula used here.
Classic checks include edge one, which yields circumradius about zero point nine five one one, and edge two, which scales linearly to about one point nine zero two one. Because the path uses only multiplication and a fixed coefficient built from square root of five, it is deterministic, O of one, and free of iterative solvers or mesh libraries.
Where icosahedron circumradius calculations show up in real work
Secondary-school and college solid-geometry courses often ask only for the circumscribed sphere radius of a regular icosahedron given the edge, separate from volume or surface-area items on the same worksheet; a trusted calculator with a strict positive-edge gate speeds homework grading and removes spreadsheet mistakes where someone confuses circumradius with inradius or multiplies by the wrong nested radical of five.
Architecture, jewelry, and manufacturing workflows size icosahedral crystals, decorative beads, geodesic-dome scale models, and Platonic-solid prints from a measured edge; the circumradius feeds clearance, packing, and circumscribed-sphere offsets while the multi-metric sibling calculator remains available when volume and surface area are needed on another page. Game, graphics, and CAD libraries occasionally store regular icosahedral cells by edge length for maps, strategy boards, and procedural meshes; deterministic circumradius supports collision proxies, bounding spheres, and layout math without a full computational-geometry stack.
Survey and sketch tools sometimes record regular icosahedral solids when only the edge is measured; having a shared numeric oracle keeps field notes and office scripts aligned. If you later need volume, surface area, inradius together, irregular icosahedra from vertices, or a different Platonic solid, treat those as separate search intents rather than overloading this circumradius-focused landing.
What you can do with it
Homework and exam icosahedron circumradius drills
Students and graders confirm the circumscribed sphere radius from a single positive edge, including clear rejection when the edge is negative or zero.
Crystal, geodesic, and model sizing from edge length
From a measured regular-icosahedron edge, recover the circumradius for clearance, packing, and bounding-sphere offsets without a full CAD stack.
FAQ
What formula does the icosahedron circumradius calculator use?
For a regular icosahedron with edge length a, the circumscribed sphere radius is R = (a/4)·√(10+2√5). The same closed form appears in standard solid-geometry references.
What happens if the edge is negative?
A regular icosahedron requires a strictly positive edge; negative values are rejected and those calls are not charged.
Is the icosahedron circumradius calculator free?
The browser tool runs free on your device.
How is this different from the general icosahedron calculator?
This landing focuses only on circumradius from edge length for the icosahedron-circumradius search intent. The multi-metric geom.icosahedron capability returns volume and surface area on a separate page.
How does the circumradius change when the edge changes?
It grows in direct proportion to the edge. An edge of 1 gives about 0.9511, an edge of 2 about 1.9021 and an edge of 6 about 5.7063, so doubling the edge doubles the radius of the sphere through the vertices.
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/icosahedron-circumradius \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/icosahedron-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/icosahedron-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/icosahedron-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/icosahedron-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.icosahedron_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 icosahedron circumradius calculator accepts a strictly positive edge, rejects negative and non-finite values with a clear invalid_input error, and returns the circumradius as structured JSON.
What the circumradius of a regular icosahedron means and how the formula works
Huge magnitudes can still overflow IEEE-754 double range; when the result is non-finite the request fails with a clear message instead of returning Infinity in JSON. This closed form matches analytic solid-geometry references and keeps automated graders aligned with hand computation on the same edge length without floating-point drift from multi-step vertex constructions or numerical root finding.
Input contract, response shape, and error behaviour
A negative edge is rejected with a message that edge must not be negative; zero is also rejected because a regular icosahedron requires a strictly positive edge. Failed validation is not charged. Example checks: edge two yields circumradius about one point nine zero two one; edge six yields about five point seven zero six three; edge negative three fails naming the negative edge. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation without reimplementing the nested-radical formula in every language. Keep requests small, pin the rounded circumradius in tests, and you get reproducible radii for lessons, LMS plugins, and edge workers without standing up a full geometry stack of your own for every client. POST a JSON body with one required field named edge: the common side length of the regular icosahedron, which must be a strictly positive finite number. The value may arrive as a JSON number or as a numeric string that parses cleanly. Missing fields, empty strings, non-numeric text, NaN, and Infinity are rejected with invalid_input naming the field. On success the JSON response echoes the edge rounded for stable goldens, returns circumradius as the circumscribed sphere radius, includes a short formula string R equals a over four root of ten plus two root five for display next to the answer, and sets method to regular_icosahedron_circumradius so logs and teaching UIs can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where icosahedron circumradius calculations show up in real work
Because this capability is a pure parse engine, failed validation does not consume balance and successful runs stay at $0.002 per item.
Integration use cases
Deterministic fixtures for solid-geometry tests
Pipelines that seed unit tests or interactive Platonic-solid demos can call a pure circumradius oracle instead of reimplementing nested-radical arithmetic in every language.
Technical questions
How do I compute the circumradius of a regular icosahedron?
Send a JSON body with a positive edge field. The response returns edge, circumradius, formula R = (a/4)·√(10+2√5), and method regular_icosahedron_circumradius.