Tetrahedron midradius calculator
Get the midradius of a regular tetrahedron, the radius of the sphere that grazes each of its six edges at the midpoint: an edge of 2 gives √2/2, about 0.707106781187. That sphere sits between the inner and the outer one. 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
- 0.7071067812
- Midsphere radius
- 0.7071067812
The midradius of a regular tetrahedron 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).
What the midradius and midsphere 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 proportional to square root of six over twelve times the edge.
The circumsphere passes through all four vertices with circumradius three times the inradius. Between them sits the midsphere, also called the intersphere or middle sphere: it is tangent to all six 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 square root of two) divided by four, which is algebraically identical to a divided by (two times 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 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.
Where the midsphere radius helps
Classrooms and tutoring platforms use a tetrahedron midradius calculator so learners can check the midsphere radius for arbitrary positive edge lengths without mixing up inradius 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 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√2/4 against textbook worked examples without confusing inradius or circumradius.
CAD and packing notebooks
Compute the edge-tangent midsphere radius for regular tetrahedral 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 tetrahedron: the radius of the sphere tangent to all six 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, surface area, height, and insphere radius from edge length, use the regular tetrahedron volume calculator (geom.tetrahedron).
What is the midradius of a regular tetrahedron with an edge of 2?
It is √2/2, about 0.707106781187, because the edge times √2 divided by four simplifies to that value. With an edge of 1 you get √2/4.
Is the edge times √2 over four the same as the edge over two times √2?
Yes. Rationalising one form gives the other, so both describe the same length. Use whichever matches your textbook; the result is identical.
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-midradius \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/tetrahedron-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/tetrahedron-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/tetrahedron-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/tetrahedron-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.tetrahedron_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 tetrahedron midradius calculator accepts one edge length, applies the closed formula ρ equals a times square root of two 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 tetrahedron
Reporting both midradius and midsphere_radius as the same field pair matches textbook language (midradius) and CAD or packing language (midsphere radius) without forcing the client to rename keys.
Input contract, response shape, and error behaviour
Example: edge equals two yields midradius equal to square root of two over two, approximately zero point seven zero seven one, because (two times square root of two) over four simplifies to square root of two over two. Edge equals one yields (square root of two) over four. 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. 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, midradius, midsphere_radius (identical to midradius), method set to regular_tetrahedron_midradius, and formula string ρ = (a · √2) / 4 for display next to the answer. Pair the response with your own renderer if you need a 3D sketch of the midsphere inside the tetrahedron; this endpoint returns scalars only and does not emit meshes, unit conversions, or volume.
Where a tetrahedron midradius API earns its keep
CAD and digital-fabrication notebooks often size edge-tangent spheres or clearance shells around tetrahedral trusses; calling a tiny parse API avoids pulling a computer-algebra system for one multiply.
Technical questions
How do I calculate the midradius of a regular tetrahedron?
Send a finite non-negative edge length as edge (or a, side, edge_length, length). The API returns midradius = (edge · √2) / 4 and the same value as midsphere_radius.
What does the tetrahedron 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.