Rhombicosidodecahedron calculator
Enter the edge length of a rhombicosidodecahedron and get its exact volume and total surface area: an edge of 1 gives a volume of about 41.6153237825 and a surface of about 59.3059828449. 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
- 41.6153237825
- Surface area
- 59.3059828449
- Formula volume
- V = (1/3)(60 + 29√5) a³
- Formula surface area
- A = (30 + 5√3 + 3√(25 + 10√5)) a²
This calculator turns a single positive edge length into the two classic metrics of a uniform rhombicosidodecahedron: the exact closed-form volume one-third times (60 + 29 times the square root of five) times a cubed, and the total surface area (30 + 5 times the square root of three + 3 times the square root of (25 + 10 times the square root of five)) times a squared.
What a rhombicosidodecahedron calculator returns and why edge length is enough
A uniform rhombicosidodecahedron is one of the thirteen Archimedean solids: twenty equilateral triangular faces, thirty square faces, and twelve regular pentagonal faces, with sixty vertices where one triangle, two squares, and one pentagon meet in cyclic order, and one hundred twenty equal edges. Because every face is a regular polygon of the same edge length and the vertex figure is fixed by symmetry, a single positive edge length a completely determines the solid up to rigid motion.
This tool exists to turn that one length into a deterministic response without forcing you to look up which closed form multiplies one-third of (60 + 29 times the square root of five) by a cubed, how the total surface relates to twenty triangle areas plus thirty square areas plus twelve pentagon areas, or whether surface area means one face or the whole net.
If you need a regular dodecahedron, icosidodecahedron, or truncated cuboctahedron alone, use a tool dedicated to those problems; the search intent here is the rhombicosidodecahedron from edge length and a clear answer or an honest error when the edge is not positive.
What you can do with it
Homework and Archimedean-solid drills
Compute rhombicosidodecahedron volume and total surface from textbook edge length, and verify that negative edges are rejected.
3D models, packaging, and game assets
Size near-spherical polyhedral containers, print nets, or procedural meshes from a known edge in CAD, manufacturing, and simulation tools.
FAQ
What does the rhombicosidodecahedron calculator cost?
It is free in your browser on this page.
Which formulas are used?
Volume is V = (1/3)(60 + 29√5) a³ and total surface area is A = (30 + 5√3 + 3√(25 + 10√5)) a², the standard closed forms for a uniform rhombicosidodecahedron of edge a.
What faces does a rhombicosidodecahedron have?
It has twenty equilateral triangles, thirty squares and twelve regular pentagons. At each of its 60 corners one triangle, two squares and one pentagon meet, always in the same order.
Does the surface area mean one face or the whole solid?
The whole solid: it adds up all sixty-two faces, the twenty triangles, thirty squares and twelve pentagons, each with the same edge length.
What happens if the edge is zero or negative?
The calculator stops with an error instead of returning a meaningless result. A zero, negative, missing or non-numeric edge does not describe a real solid.
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/rhombicosidodecahedron \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":1}'const res = await fetch("https://api.kit.forhosting.com/geom/rhombicosidodecahedron", {
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/rhombicosidodecahedron",
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/rhombicosidodecahedron", 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/rhombicosidodecahedron", 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.rhombicosidodecahedron",
"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
You enter the edge as a finite positive length; the engine returns edge, volume, surface_area, and formula strings so a student or a log line can verify the identity without reading the source. When the edge is missing, zero, negative, or non-finite, the tool returns invalid_input instead of a silent NaN. Through the API a successful request costs $0.002; validation failures are not charged.
What a rhombicosidodecahedron calculator returns and why edge length is enough
The response always echoes edge, reports volume from the standard Archimedean formula, surface_area as the total of all sixty-two faces, and formula strings for both so a student or a log line can verify the identity without reading the source. Missing edge, non-finite values, zero length, or a negative length all raise invalid_input with a message that names the failing field. That discipline keeps automated tests stable, makes caching safe, and prevents a script from treating a degenerate segment as a real solid.
How the closed forms are evaluated: volume and total surface area
Volume uses the exact Archimedean expression volume equals one-third times open parenthesis sixty plus twenty-nine times square root of five close parenthesis, times edge cubed. Total surface area multiplies thirty plus five times square root of three plus three times the square root of twenty-five plus ten times square root of five, by edge squared; equivalently it is twenty times the area of an equilateral triangle of side a, plus thirty times a squared for the squares, plus twelve times the area of a regular pentagon of side a, written as a compact radical that avoids intermediate face-area rounding. Optional precision only rounds the published fields; the internal solve always uses full double precision before that final rounding step. The engine resolves edge from the field edge with aliases a, edge_length, and side, requiring the value to be finite and strictly greater than zero. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the radical coefficients yourself.
Where rhombicosidodecahedron metrics fit in study, design, and code
Students meet the rhombicosidodecahedron when a textbook asks for volume from edge length, when an exam demands total surface of an Archimedean solid, or when solid-geometry homework wants material estimates for a sixty-two-faced model before painting or 3D printing. Teachers can pre-generate answer keys that list volume and surface for integer edges, and that refuse zero or negative lengths instead of printing NaN. Mechanical and product design sometimes needs the capacity of a near-spherical polyhedral container, the sheet area of a folded net, or a sanity check against a CAD volume property; knowing that a requested edge is not positive is as valuable as the number itself. Game and simulation teams use volume and surface for collision approximations, particle bounds, and procedural mesh sizing that should stay deterministic across platforms. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split—free interactive exploration and a priced, identical API—matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with other free geometry endpoints when you need dodecahedron, cuboctahedron, or torus solvers for a lab of deterministic tools, and with documentation that shows both the edge and the finished metrics so the numbers and the picture stay aligned for learners and reviewers. Whether you arrived searching for a rhombicosidodecahedron calculator or for Archimedean volume and surface from edge length, the contract is the same: positive edge in, finite metrics or a clear invalid_input out.
Integration use cases
Deterministic geometry fixtures
Generate reproducible Archimedean-solid metrics for unit tests, contest validators, and CI assertions without hand-rolled radical coefficients.
Technical questions
What if the edge length is negative?
The handler returns invalid_input with a message that the edge must be a positive finite number; negative edges are invalid.
Does surface_area mean one face or the whole solid?
surface_area is the total surface of all sixty-two faces: twenty equilateral triangles, thirty squares, and twelve regular pentagons. It is not a single face area.
Can I control rounding?
Yes. Set precision (aliases decimals, dp) to an integer from 0 to 15; the default is 10 decimal places on all numeric fields.