Rhombic triacontahedron calculator
Enter the edge length of a rhombic triacontahedron, the 30-face Catalan solid behind many zomes and decorative shells, and get its volume and surface area: an edge of 1 gives about 12.310734148701 and 26.832815729997. 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
- 12.3107341487
- Surface area
- 26.83281573
- Faces
- 30
- Edges
- 60
Students, jewelers, 3D designers, and exam writers need its volume and total surface area from that shared edge length when sizing quasi-spherical shells, checking Catalan solid tables, or scripting parametric models for zomes and decorative polyhedra. This rhombic triacontahedron calculator accepts one edge length, applies the closed formulas V equals four times the square root of five plus two times the square root of five times a cubed and S equals twelve times the square root of five times a squared, and returns both metrics with the solid’s fixed face, edge, and vertex counts.
What a rhombic triacontahedron is and which formulas fix volume and surface
Start from the dual of the icosidodecahedron: place a vertex at the center of each triangular and pentagonal face of that Archimedean solid and connect vertices that share an edge between faces. Because every edge has the same length a after a uniform dual construction, 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 four times the square root of five plus two times the square root of five, times a cubed, and total surface area S equals twelve times the square root of five times a squared. 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.
What you can do with it
CAD, jewelry, and parametric design notebooks
Compute volume and surface area for rhombic triacontahedron cells in scripts that size decorative polyhedra, zome shells, or clearance envelopes.
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
What happens if the edge is negative?
A rhombic triacontahedron requires a non-negative edge length; negative values are rejected and those calls are not charged.
Is the rhombic triacontahedron calculator free?
The browser tool runs free on your device.
How does a rhombic triacontahedron differ from an icosidodecahedron?
An icosidodecahedron is Archimedean (vertex-transitive) with twenty triangles and twelve pentagons.
Which formulas does the calculator use?
The volume is four times the square root of five plus twice the square root of five, times the edge cubed. The surface area is twelve times the square root of five, times the edge squared.
What happens to the results if I double the edge?
The volume grows eight times and the surface area four times, because the volume depends on the edge cubed and the area on the edge squared.
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/rhombic-triacontahedron \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/rhombic-triacontahedron", {
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/rhombic-triacontahedron",
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/rhombic-triacontahedron", 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/rhombic-triacontahedron", 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.rhombic_triacontahedron",
"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 rhombic triacontahedron 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 twelve point three one one and surface area approximately twenty-six point eight three three 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 common edge length of the rhombic triacontahedron. 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 thirty, edges set to sixty, vertices set to thirty-two, method set to rhombic_triacontahedron, and two formula strings for display next to the answer: V = 4 · √(5 + 2√5) · a³ and S = 12 · √5 · a². Pair the response with your own renderer if you need a 3D mesh of the rhombic triacontahedron; this endpoint returns scalars and topology only and does not emit meshes, unit conversions, or midradius or circumradius.
Where a rhombic triacontahedron calculator API earns its keep
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 dodecahedron, icosahedron, or icosidodecahedron calculator, this landing targets the narrower search intent of the rhombic triacontahedron alone, so SEO pages do not compete for ordinary Platonic or Archimedean solid volume queries while still serving the exact scalars designers and students ask for when they type rhombic triacontahedron into a search box and need volume plus surface area from edge length. CAD, digital-fabrication, jewelry, and parametric-design notebooks often size Catalan cells, zome-related 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 rhombic triacontahedron metrics
Let students submit an edge length and verify volume V=4√(5+2√5)·a³ and surface S=12√5·a² against textbook Catalan solid tables.
Technical questions
How do I compute rhombic triacontahedron 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.
Which formulas does the calculator use?
Volume V = 4 · √(5 + 2√5) · a³. Surface area S = 12 · √5 · a², where a is the common edge length of the rhombic triacontahedron.