Truncated octahedron calculator
From one edge length you get the volume and surface area of a truncated octahedron: an edge of 1 means a volume of 11.313708498985 and a surface of 26.784609690827. Useful when sizing space-filling cells or checking a textbook table. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Edge
- 1
- Volume
- 11.313708499
- Surface area
- 26.7846096908
- Faces
- 14
- Edges
- 36
A truncated octahedron is the Archimedean solid obtained by cutting every vertex of a regular octahedron until the original triangular faces become regular hexagons and squares appear where the vertices used to be. Students, 3D designers, crystallographers, and exam writers need its volume and total surface area from a single shared edge length when sizing space-filling cells, checking textbook answers, or scripting parametric models. This truncated octahedron calculator accepts one edge length, applies the closed formulas V equals eight times the square root of two times a cubed and S equals six plus twelve times the square root of three times a squared, and returns both metrics with the solid’s fixed face, edge, and vertex counts.
What a truncated octahedron is and which formulas fix volume and surface
Start from a regular octahedron and truncate (cut off) each of the six vertices by the same depth until the original equilateral triangular faces become regular hexagons and new square faces appear at the former vertices. The same solid also arises by truncating the vertices of a cube until its square faces become regular hexagons; both constructions yield the Archimedean truncated octahedron. The resulting convex polyhedron is one of the thirteen Archimedean solids: all faces are regular polygons, and the vertex figure is the same at every vertex.
It has fourteen faces in total—eight regular hexagons and six squares—thirty-six equal edges, and twenty-four identical vertices. Because every edge has the same length a after a uniform truncation, a single non-negative length completely determines the solid up to congruence. The classical closed formulas used in solid-geometry tables are volume V equals eight times the square root of two times a cubed, and total surface area S equals the quantity six plus twelve times the square root of three times a squared.
Those expressions come from summing the areas of eight regular hexagons of side a and six squares of side a, plus the known volume of the fully truncated octahedron in terms of edge length. When a is zero the solid collapses and both volume and surface area are zero; when a is positive both grow as a cubed and a squared respectively. Negative a has no interpretation as a geometric edge length, so this capability refuses it rather than taking an absolute value that would hide a data-entry bug.
Where truncated octahedron figures are useful
Classrooms and tutoring platforms use a truncated octahedron calculator so learners can check volume and surface area for arbitrary non-negative edge lengths without mixing the hexagon and square face formulas by hand. Exam-prep generators need a deterministic oracle that marks negative edges as invalid input instead of silently reflecting the absolute value or inventing a positive volume. Failed validation does not consume balance.
Compared with a plain octahedron calculator, this landing targets the narrower search intent of the truncated octahedron alone, so SEO pages do not compete for ordinary octahedron volume queries while still serving the exact scalars designers and students ask for when they type truncated octahedron into a search box and need volume plus surface area from edge length.
What you can do with it
CAD and space-filling cell design
Compute volume and surface area for truncated-octahedron cells in scripts that size honeycombs, packaging shells, or clearance envelopes that tile 3D space.
Exam generators and LMS auto-grading
Mark solid-geometry items with a deterministic oracle that rejects negative edges and pins twelve-decimal goldens for volume and surface area.
FAQ
Which formulas does the calculator use?
Surface area S = (6 + 12√3) · a², where a is the common edge length of the truncated octahedron.
What happens if the edge is negative?
A truncated octahedron requires a non-negative edge length; negative values are rejected and those calls are not charged.
Is the truncated octahedron calculator free?
The browser tool runs free on your device.
How does a truncated octahedron differ from a regular octahedron?
A regular octahedron has eight equilateral triangular faces. A truncated octahedron has eight regular hexagons and six squares (14 faces, 36 edges, 24 vertices) after cutting off the octahedron’s vertices uniformly.
What happens when the edge is zero?
The solid collapses, so both volume and surface area come out as zero. Only a negative edge is refused, because it cannot be a real length and 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/truncated-octahedron \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/truncated-octahedron", {
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/truncated-octahedron",
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/truncated-octahedron", 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/truncated-octahedron", 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.truncated_octahedron",
"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
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 a truncated octahedron is and which formulas fix volume and surface
Reporting volume, surface_area, and the topology triple faces, edges, vertices matches textbook language and CAD notebooks without forcing clients to re-derive face counts from memory.
Input contract, response shape, and error behaviour
Example: edge equals one yields volume approximately eleven point three one four and surface area approximately twenty-six point seven eight five in consistent cubic and square units. Edge equals two scales volume by eight and surface area by four because of the a cubed and a squared dependence. Keep requests small, pin the rounded volume and surface area in automated tests, and you get reproducible Archimedean metrics 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 truncated octahedron. 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, volume, surface_area, faces set to fourteen, edges set to thirty-six, vertices set to twenty-four, method set to truncated_octahedron, and two formula strings for display next to the answer: V = 8√2 · a³ and S = (6 + 12√3) · a². Pair the response with your own renderer if you need a 3D mesh of the truncated octahedron; this endpoint returns scalars and topology only and does not emit meshes, unit conversions, or midradius or circumradius.
Where a truncated octahedron calculator API earns its keep
CAD, digital-fabrication, materials-science, and parametric-design notebooks often size Archimedean space-filling cells—the truncated octahedron tiles Euclidean three-space by translation alone—packaging shells, or decorative polyhedra; calling a tiny parse API avoids pulling a computer-algebra system for two closed multiplications.
Integration use cases
Classroom checks of truncated-octahedron metrics
Let students submit an edge length and verify volume V=8√2·a³ and surface S=(6+12√3)·a² against textbook Archimedean solid tables.
Technical questions
How do I compute truncated octahedron volume and surface area?
Send a finite non-negative edge length as edge (aliases a, side, edge_length, length). The API returns volume, surface_area, faces, edges, vertices, and the closed formulas used.