Tetrahedron inradius calculator
Find the radius of the largest ball that fits inside a regular tetrahedron and touches all four faces: an edge of 2 gives √6/6, about 0.408248290464. A zero edge gives zero, negative edges are rejected. Free in your browser.
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Run — free
Worked example
- Edge
- 2
- Inradius
- 0.4082482905
- Insphere radius
- 0.4082482905
The inradius of a regular tetrahedron is the radius of the insphere—the unique sphere that lies inside the solid and touches every face at its centroid. Students, CAD users, and exam writers often need that single length when packing, dimensioning clearances, or comparing the three classical spheres (in-, mid-, and circumsphere).
What the inradius and insphere mean for a regular tetrahedron
A regular tetrahedron has four equilateral triangular faces, six equal edges, and four vertices. Three concentric spheres are classically associated with that solid. The insphere is tangent to the four faces and has inradius r equal to a times square root of six divided by twelve.
The circumsphere passes through all four vertices with circumradius three times the inradius, equal to a times square root of six over four. Between them sits the midsphere, tangent to all six edges, with midradius a times square root of two over four. This capability isolates the face-tangent case only: given a common edge length a, it returns the inradius of the inscribed sphere.
That value is independent of orientation—only the common edge length matters. When a is zero the solid collapses and the inradius is zero; when a is positive the insphere 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 insphere radius is used
Classrooms and tutoring platforms use a tetrahedron inradius calculator so learners can check the inscribed sphere radius for arbitrary non-negative edge lengths without mixing up midradius and circumradius formulas. 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-tetrahedron volume tool, this landing targets the narrower search intent of inradius and insphere alone, so SEO pages do not compete for volume or midradius queries while still serving the exact scalar designers and students ask for when they type inradius or inscribed sphere into a search box.
What you can do with it
Classroom checks of insphere radius
Let students submit an edge length and verify the inradius r = a√6/12 against textbook worked examples without confusing midradius or circumradius.
CAD and packing notebooks
Compute the face-tangent insphere radius for regular tetrahedral cells in scripts that size inner clearance shells or packing spheres.
Exam and LMS auto-grading
Mark inradius 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 inradius and insphere radius?
They are the same geometric length for a regular tetrahedron: the radius of the sphere tangent to all four faces.
What happens if the edge is negative?
A negative edge length is not a valid geometric input and is not billed.
What is the inradius of a regular tetrahedron with an edge of 2?
It is √6/6, about 0.408248290464, because the edge times √6 divided by twelve simplifies to that value. The sphere of that radius touches each face at its centre.
How does the inradius compare with the circumradius?
The sphere through the four vertices has a radius three times larger, the edge times √6 over four. The sphere touching the six edges sits in between, with the edge times √2 over four.
Where is the insphere radius used in practice?
In CAD and packing work, to size an inner clearance shell or a packing sphere inside tetrahedral cells, and in classrooms to check textbook examples without mixing up the three sphere formulas.
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/tetrahedron-inradius \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/tetrahedron-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/tetrahedron-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/tetrahedron-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/tetrahedron-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.tetrahedron_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 tetrahedron inradius calculator accepts one edge length, applies the closed formula r equals a times square root of six over twelve, and returns both inradius and insphere_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 inradius and insphere mean for a regular tetrahedron
Reporting both inradius and insphere_radius as the same field pair matches textbook language (inradius) and CAD or packing language (insphere radius) without forcing the client to rename keys.
Input contract, response shape, and error behaviour
Example: edge equals two yields inradius equal to square root of six over six, approximately zero point four zero eight two, because (two times square root of six) over twelve simplifies to square root of six over six. Edge equals one yields (square root of six) over twelve. Keep requests small, pin the rounded inradius in automated tests, and you get reproducible packing radii for classroom checks, CAD notebooks, and production pipelines without standing up a full solid-geometry library or 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 tetrahedron. 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, inradius, insphere_radius (identical to inradius), method set to regular_tetrahedron_inradius, and formula string r = (a · √6) / 12 for display next to the answer. Pair the response with your own renderer if you need a 3D sketch of the insphere inside the tetrahedron; this endpoint returns scalars only and does not emit meshes, unit conversions, or volume.
Where a tetrahedron inradius API earns its keep
CAD and digital-fabrication notebooks often size face-tangent spheres or inner clearance shells inside tetrahedral cells; calling a tiny parse API avoids pulling a computer-algebra system for one multiply by square root of six over twelve.
Technical questions
How do I calculate the inradius of a regular tetrahedron?
Send a finite non-negative edge length as edge (or a, side, edge_length, length). The API returns inradius = (edge · √6) / 12 and the same value as insphere_radius.
Does this return volume or midradius as well?
No. This capability focuses on inradius only. For volume, surface area, height, and multi-metric output use geom.tetrahedron; for midsphere radius use geom.tetrahedron_midradius.
What does the tetrahedron inradius calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.