Icosahedron midradius calculator
An icosahedron with an edge of 2 has a midradius of about 1.618: that is the radius of the sphere touching all thirty edges at their midpoints. Enter any edge and get it from a(1+√5)/4. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Edge
- 2
- Midradius
- 1.6180339888
- Midsphere radius
- 1.6180339888
The midradius of a regular icosahedron is the radius of the midsphere—the unique sphere that touches every edge of the solid. Students, CAD users, and exam writers often need that single length when packing, dimensioning, or comparing the three classical spheres (in-, mid-, and circumsphere) of the regular solid with twenty triangular faces.
What the midradius and midsphere mean for a regular icosahedron
A regular icosahedron has twenty equilateral triangular faces, twelve vertices, and thirty equal edges. Three concentric spheres are classically associated with that Platonic solid. The insphere is tangent to the twenty faces and has inradius proportional to three times square root of three plus square root of fifteen over twelve times the edge.
The circumsphere passes through all twelve vertices with circumradius equal to one quarter times square root of ten plus two times square root of five times the edge. Between them sits the midsphere, also called the intersphere or middle sphere: it is tangent to all thirty edges at their midpoints in the regular case, and its radius is the midradius ρ. For edge length a the closed formula is ρ equals a times one plus square root of five, all divided by four.
The value is independent of orientation: only the common edge length matters. When a is zero the solid collapses and the midradius is zero; when a is positive the midsphere 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.
Who needs the icosahedron midradius
Classrooms and tutoring platforms use an icosahedron midradius calculator so learners can check the midsphere radius for arbitrary positive edge lengths without mixing up inradius and circumradius formulas or confusing the icosahedron with the dual dodecahedron. Exam-prep generators need a deterministic oracle that marks negative edges as invalid input instead of silently reflecting the absolute value. Failed validation does not consume balance.
Compared with a general regular-icosahedron volume tool, this landing targets the narrower search intent of midradius and midsphere alone, so SEO pages do not compete for volume or surface-area queries while still serving the exact scalar designers and students ask for when they type midradius into a search box.
What you can do with it
Classroom checks of midsphere radius
Let students submit an edge length and verify the midradius ρ = a(1+√5)/4 against textbook worked examples without confusing inradius or circumradius.
CAD and packing notebooks
Compute the edge-tangent midsphere radius for regular icosahedral cells in scripts that size clearance shells or packing spheres.
Exam and LMS auto-grading
Mark midradius items with a deterministic oracle that rejects negative edges and returns a stable twelve-decimal answer for positive edges.
FAQ
What is the difference between midradius and midsphere radius?
They are the same geometric length for a regular icosahedron: the radius of the sphere tangent to all thirty edges.
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 midradius only. For volume and surface area from edge length, use the regular icosahedron calculator (geom.icosahedron).
Where does the midsphere sit compared with the insphere and the circumsphere?
Between the two. The insphere touches the twenty faces, the circumsphere passes through the 12 vertices, and the midsphere touches the thirty edges at their midpoints, so its radius falls between the inradius and the circumradius.
What happens with an edge of zero?
The solid collapses to a point and the midradius is simply zero. Only a negative edge is refused, because a length cannot be negative and quietly taking its absolute value would hide a typing mistake.
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-midradius \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/icosahedron-midradius", {
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-midradius",
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-midradius", 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-midradius", 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_midradius",
"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 midradius calculator accepts one edge length, applies the closed formula ρ equals a times one plus square root of five over four, and returns both midradius and midsphere_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 midradius and midsphere mean for a regular icosahedron
Reporting both midradius and midsphere_radius as the same field pair matches textbook language and CAD or packing language without forcing the client to rename keys.
Input contract, response shape, and error behaviour
Edge equals one yields one plus square root of five over four, approximately zero point eight zero nine. Keep requests small, pin the rounded midradius in automated tests, and you get reproducible packing radii for homework, LMS plugins, and edge workers without standing up a full solid-geometry library or a computer-algebra system. POST a JSON body with one required field edge: a finite number or numeric string for the common edge length of the regular icosahedron. 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, midradius, midsphere_radius identical to midradius, method set to regular_icosahedron_midradius, and formula string ρ equals a times one plus square root of five over four for display next to the answer. Pair the response with your own renderer if you need a three-dimensional sketch of the midsphere inside the icosahedron; this endpoint returns scalars only and does not emit meshes, unit conversions, volume, or surface area.
Technical questions
How do I calculate the midradius of a regular icosahedron?
Send a finite non-negative edge length as edge (or a, side, edge_length, length). The API returns midradius = (edge · (1 + √5)) / 4 and the same value as midsphere_radius.
What does the icosahedron midradius calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.