Triakis tetrahedron calculator
Get the volume and surface of a triakis tetrahedron, the twelve-faced solid made by raising a small pyramid on each face of a tetrahedron: a shorter edge of 1 gives a volume of 0.982092751648 and a surface of 5.527707983926. 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
- 0.9820927516
- Surface area
- 5.5277079839
- Faces
- 12
- Edges
- 18
A triakis tetrahedron is the Catalan solid dual to the truncated tetrahedron: start from a regular tetrahedron and attach matching triangular pyramids to each face so the solid becomes a twelve-faced polyhedron whose faces are congruent isosceles triangles. Students, mineralogists (who may know it as a tristetrahedron), and 3D designers need its volume and total surface area from a single shorter edge length when checking Catalan tables, sizing dual Archimedean cells, or scripting parametric models. This triakis tetrahedron calculator accepts one non-negative edge length, applies the closed MathWorld forms volume equals twenty-five over thirty-six times the square root of two times a cubed and surface area equals five over three times the square root of eleven times a squared, then returns both metrics with the solid’s fixed face, edge, and vertex counts.
What a triakis tetrahedron is and which formulas fix volume and surface
Begin with a regular tetrahedron and perform a kis operation: attach a triangular pyramid to each of the four faces so that every original face is replaced by three congruent isosceles triangles meeting at a new apex above that face. The resulting convex polyhedron is the triakis tetrahedron, the Catalan dual of the truncated tetrahedron (an Archimedean solid).
It has twelve identical isosceles triangular faces, eighteen edges of two lengths (six shorter edges corresponding to the MathWorld parameter s1 and twelve longer face edges), and eight vertices (four of degree three and four of degree six). A single non-negative shorter edge length a completely determines the metrics up to congruence when the face proportions are fixed by the dual of a unit-edge truncated tetrahedron.
MathWorld normalizes so that s1 equals a and publishes the closed surface and volume formulas S equals five thirds times the square root of eleven times a squared, and V equals twenty-five over thirty-six times the square root of two times a cubed. 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.
Who uses triakis tetrahedron measurements
Classrooms and tutoring platforms use a triakis tetrahedron calculator so learners can check volume and surface area for arbitrary non-negative shorter edge lengths without mixing Catalan dual formulas with regular tetrahedron tables or confusing s1 with the longer face edges. 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 tetrahedron or dodecahedron calculator, this landing targets the narrower search intent of the triakis tetrahedron alone, so SEO pages do not compete for ordinary Platonic solid volume queries while still serving the exact scalars designers and students ask for when they type triakis tetrahedron into a search box and need volume plus surface area from edge length.
What you can do with it
Classroom checks of triakis tetrahedron metrics
Let students submit a shorter edge length and verify V=(25/36)·√2·a³ and S=(5/3)·√11·a² against Catalan solid tables, plus topology 12 faces, 18 edges, 8 vertices.
CAD and crystal-model notebooks
Compute volume and surface area for triakis tetrahedron (tristetrahedron) cells in scripts that size dual Archimedean solids or mineralogical display models.
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?
With shorter edge a (MathWorld s1): surface area S = (5/3)·√11·a² and volume V = (25/36)·√2·a³. Topology is fixed at 12 faces, 18 edges, and 8 vertices.
What happens if the edge is negative?
A triakis tetrahedron requires a non-negative edge length; negative values are rejected and those calls are not charged.
Is the triakis tetrahedron calculator free?
The browser tool runs free on your device.
How does a triakis tetrahedron differ from a regular tetrahedron?
A regular tetrahedron has four equilateral faces and volume (√2/12)·a³. A triakis tetrahedron is a Catalan solid with twelve isosceles triangular faces (dual of the truncated tetrahedron) and uses V=(25/36)·√2·a³ for shorter edge a.
Which edge of a triakis tetrahedron should I enter?
Enter the shorter edge. The solid has 6 shorter edges and 12 longer ones on its isosceles faces, and the formulas are written for the shorter length.
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/triakis-tetrahedron \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":1}'const res = await fetch("https://api.kit.forhosting.com/geom/triakis-tetrahedron", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"edge": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/triakis-tetrahedron",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"edge": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/triakis-tetrahedron", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"edge":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"edge":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/triakis-tetrahedron", 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": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.triakis_tetrahedron",
"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 triakis tetrahedron 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 or to confuse this solid with a regular tetrahedron, which has only four equilateral faces and a different volume formula.
Input contract, response shape, and error behaviour
Example: edge equals one yields volume approximately zero point nine eight two and surface area approximately five point five two eight 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 Catalan 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 shorter edge length of the triakis tetrahedron under the MathWorld s1 normalization. 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 twelve, edges set to eighteen, vertices set to eight, method set to triakis_tetrahedron, and two formula strings for display next to the answer: V equals open parenthesis twenty-five over thirty-six close parenthesis times square root of two times a cubed, and S equals open parenthesis five over three close parenthesis times square root of eleven times a squared, both with a as the shorter edge. Pair the response with your own renderer if you need a 3D mesh or net of the dual truncated tetrahedron; this endpoint returns scalars and topology only and does not emit meshes, unit conversions, midradius, inradius, or the longer face-edge length as a separate free parameter.
Where a triakis tetrahedron calculator API earns its keep
CAD, digital-fabrication, and parametric-design notebooks often size dual Archimedean cells, mineralogical crystal models labeled tristetrahedron, or decorative Catalan polyhedra; calling a tiny parse API avoids pulling a computer-algebra system for two multiplications after square roots of two and eleven.
Technical questions
How do I compute triakis tetrahedron volume and surface area?
Send a finite non-negative shorter edge length as edge (aliases a, side, edge_length, length). The API returns volume, surface_area, faces, edges, vertices, and the closed formulas used.