Dodecahedron inradius calculator
Enter the edge of a regular dodecahedron and get the radius of the largest sphere that fits inside it: an edge of 2 gives an inradius of about 2.2270327288, from (a/20)·√(250 + 110√5). Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Edge
- 2
- Inradius
- 2.2270327288
A regular dodecahedron is a Platonic solid with twelve congruent regular pentagonal faces, twenty vertices where three faces meet, and thirty equal edges. When only the common edge length is known, the radius of the sphere that sits inside the solid and touches every face—the inradius—is fixed by a closed radical form involving the square root of five: edge over twenty, times the square root of two hundred fifty plus one hundred ten times square root of five. Students checking solid-geometry homework, exam writers who need a deterministic answer key, makers sizing crystal or gem models from a measured edge, and game or CAD pipelines that store regular dodecahedra by edge length all need that single number without re-deriving nested radicals by hand.
What the inradius of a regular dodecahedron means and how the formula works
In solid geometry the inradius of a polyhedron is the radius of the largest sphere that fits entirely inside the solid and is tangent to every face. For a regular dodecahedron that sphere is uniquely determined once the common edge length is fixed. Elementary vector geometry shows that the perpendicular distance from the center to any face equals a over twenty times the square root of two hundred fifty plus one hundred ten times square root of five, which is exactly the inradius formula used here.
The same identity appears in textbooks as r equals a over twenty root of two hundred fifty plus one hundred ten root five, or equivalently in forms that factor ten and nested radicals of five. Classic checks include edge one, which yields inradius about one point one one three five, and edge two, which scales linearly to about two point two two seven zero.
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 dodecahedron inradius calculations show up in real work
Secondary-school and college solid-geometry courses often ask only for the inscribed sphere radius of a regular dodecahedron 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 inradius with circumradius or multiplies by the wrong nested radical of five.
Architecture, jewelry, and manufacturing workflows size dodecahedral crystals, decorative beads, and Platonic-solid models from a measured edge; the inradius feeds clearance, packing, and inscribed-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 dodecahedral cells by edge length for maps, strategy boards, and procedural meshes; deterministic inradius supports collision proxies and layout math without a full computational-geometry stack.
Survey and sketch tools sometimes record regular dodecahedral 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, circumradius together, irregular dodecahedra from vertices, or a different Platonic solid, treat those as separate search intents rather than overloading this inradius-focused landing.
What you can do with it
Homework and exam dodecahedron inradius drills
Students and graders confirm the inscribed sphere radius from a single positive edge, including clear rejection when the edge is negative or zero.
Crystal, jewelry, and model sizing from edge length
From a measured regular-dodecahedron edge, recover the inradius for clearance, packing, and inscribed-sphere offsets without a full CAD stack.
FAQ
What formula does the dodecahedron inradius calculator use?
For a regular dodecahedron with edge length a, the inscribed sphere radius is r = (a/20)·√(250+110√5). The same closed form appears in standard solid-geometry references.
What happens if the edge is negative?
A regular dodecahedron requires a strictly positive edge; negative values are rejected and those calls are not charged.
Is the dodecahedron inradius calculator free?
The browser tool runs free on your device.
How is this different from the general dodecahedron calculator?
This landing focuses only on inradius from edge length for the dodecahedron-inradius search intent. The multi-metric geom.dodecahedron capability returns volume and surface area on a separate page.
Is the inradius the same as the circumradius?
No. The inradius belongs to the sphere inside the solid that touches the middle of every face, while the circumradius reaches out to the corners. Mixing the two up is a common homework slip.
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/dodecahedron-inradius \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/dodecahedron-inradius", {
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/dodecahedron-inradius",
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/dodecahedron-inradius", 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/dodecahedron-inradius", 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.dodecahedron_inradius",
"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 dodecahedron inradius calculator accepts a strictly positive edge, rejects negative and non-finite values with a clear invalid_input error, and returns the inradius as structured JSON.
What the inradius of a regular dodecahedron 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 face constructions.
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 dodecahedron requires a strictly positive edge. Failed validation is not charged. Example checks: edge two yields inradius about two point two two seven zero; edge six yields about six point six eight one one; 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 inradius 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 dodecahedron, 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 inradius as the inscribed sphere radius, includes a short formula string r equals a over twenty root of two hundred fifty plus one hundred ten root five for display next to the answer, and sets method to regular_dodecahedron_inradius so logs and teaching UIs can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where dodecahedron inradius 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 inradius oracle instead of reimplementing nested-radical arithmetic in every language.
Technical questions
How do I compute the inradius of a regular dodecahedron?
Send a JSON body with a positive edge field. The response returns edge, inradius, formula r = (a/20)·√(250+110√5), and method regular_dodecahedron_inradius.