ForHosting KIT · Math

Truncated icosidodecahedron

With one edge length you get the volume and surface of a truncated icosidodecahedron: an edge of 1 gives a volume of 206.803398874989 and a surface of 174.292030342324, spread over squares, hexagons and decagons. Free in your browser.

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Worked example

If you enter

Edge: 1

You get

Volume: 206.803398875
Surface area: 174.2920303423
Square area: 1
Hexagon area: 2.5980762114

A truncated icosidodecahedron is an Archimedean solid obtained by truncating both the vertices and edges of a regular dodecahedron or icosahedron until every face is a regular polygon and every edge has the same length. The uniform solid—also known as the great rhombicosidodecahedron—has thirty square faces, twenty regular hexagonal faces, twelve regular decagonal faces, one hundred eighty equal edges, and one hundred twenty vertices. Given a single positive common edge length a, closed-form formulas answer the two measures people look up most often: volume equals ninety-five plus fifty times the square root of five, all times a cubed, and total surface area is thirty times one plus the square root of three plus the square root of five plus two square roots of five, all times a squared. Students, fabricators, CAD pre-checks, and packaging workflows all get stable numbers without installing a polyhedron library on every machine.

What volume and surface area mean for a truncated icosidodecahedron

A uniform truncated icosidodecahedron is one of the thirteen Archimedean solids: every face is a regular polygon, every edge has the same length a, and the vertex figure is the same at every corner. The solid carries thirty square faces, twenty regular hexagonal faces that arise from truncating triangular and pentagonal origins in a coordinated way, and twelve regular decagonal faces that come from the deeper truncation of pentagonal structure. Surface area is the sum of thirty unit-square areas, twenty regular-hexagon areas, and twelve regular-decagon areas.

Each square has area a squared; each regular hexagon has area three square roots of three over two times a squared; each regular decagon has area five halves times the square root of five plus two square roots of five times a squared. Multiplying and adding yields the compact closed form thirty times the quantity one plus the square root of three plus the square root of five plus two square roots of five, all times a squared.

Volume uses the classical Archimedean formula V equals ninety-five plus fifty times the square root of five times a cubed—the enclosed space of the solid when all one hundred eighty edges equal a. Units stay consistent with whatever length unit you pass for a: metres yield cubic metres of volume and square metres of area; inches yield cubic and square inches.

A classic classroom check is a equals one: volume is ninety-five plus fifty square root of five, and surface is thirty times one plus square root of three plus square root of five plus two square root of five. Edge two multiplies volume by eight and surface by four. Zero edge length is degenerate and rejected; a negative edge is never a valid geometric measure under this contract.

The arithmetic is elementary multiplications and square roots after coercion, so there is no iteration, no Monte Carlo sampling, no dependence on wall-clock time, and no need for face coordinates or dihedral angles in the input. Related solids such as a plain icosidodecahedron, an irregular truncation with unequal edges, a truncated dodecahedron, a truncated tetrahedron, or a conical frustum need different formulas and are out of scope here.

What you can do with it

Packing, takeoff, and material estimates

Turn a truncated icosidodecahedron edge into volume for fill or mass estimates, or total surface for coating, wrap, cladding, and material orders on Archimedean stock.

CAD, fabrication, and game pipeline pre-checks

Validate uniform truncated icosidodecahedron dimensions in scripts before meshing or quoting: volume for density estimates and surface for cut-lists and property panels.

FAQ

What is the formula for the volume of a truncated icosidodecahedron?

With common edge length a, volume is (95 + 50√5)·a³ for the uniform Archimedean solid. The edge must be positive and use a consistent length unit; volume is then in the matching cubic unit.

What is the surface area of a truncated icosidodecahedron?

Total surface area is 30(1 + √3 + √(5 + 2√5))·a²: thirty squares each of area a², twenty regular hexagons each of area (3√3/2)·a², and twelve regular decagons each of area (5/2)·√(5 + 2√5)·a².

How is a truncated icosidodecahedron different from a truncated dodecahedron?

A truncated dodecahedron has twelve decagons and twenty triangles. The truncated icosidodecahedron is a deeper Archimedean truncation with thirty squares, twenty hexagons, and twelve decagons (180 edges).

Why is a negative edge rejected?

A solid truncated icosidodecahedron needs a strictly positive edge.

Is the truncated icosidodecahedron calculator free?

The browser tool is free and runs locally.

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

POSThttps://api.kit.forhosting.com/geom/truncated-icosidodecahedron

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-icosidodecahedron \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"edge":2}'

const res = await fetch("https://api.kit.forhosting.com/geom/truncated-icosidodecahedron", {
  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-icosidodecahedron",
    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-icosidodecahedron", 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-icosidodecahedron", 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_icosidodecahedron",
  "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

Per request$0.002

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 capability packages those identities as deterministic JSON with no computer-algebra system, no random seeds, and the same pure solve function in the free browser tool and the prepaid API path. A negative edge is rejected with invalid_input so scripts and grading pipelines see an honest error instead of a silent absolute value or a nonsense solid.

How to call the API and what comes back

The value must be a finite number strictly greater than zero and inside the published limits up to one trillion. Use it from geometry homework pipelines, packing and takeoff scripts, interview-prep sandboxes, CAD pre-checks, fabrication cut-lists, or any workflow that needs reproducible truncated-icosidodecahedron numbers without installing a computer-algebra package on every machine. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation of many edge lengths in one session. Send a JSON body with a single required field edge—the positive common edge length of the uniform truncated icosidodecahedron. Aliases a, side, edge_length, and length are accepted and map to the same meaning. On success the response echoes edge; returns volume and surface_area as the primary results; and includes square_area for one square face, hexagon_area for one regular hexagonal face, decagon_area for one regular decagonal face, method set to truncated_icosidodecahedron, and a short formula block documenting the closed forms. Failed validation—missing the required field, a non-finite number, a negative edge, a zero edge, or a value outside published limits—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Pair the call with unit conversion only when your source data mixes millimetres and metres; this capability does not convert linear units for you, so keep the edge in one consistent unit before you send the request and interpret volume and area in the matching cubic and square units.

Practical tips, limits, and related searches

When teaching, prefer paired examples: edge one against the classic radical expressions; edge two for the eight-times volume and four-times surface scale check; and a negative edge to demonstrate the hard rejection path. Prefer this tool when you already know the uniform edge length and need volume plus total surface in one shot. Round-trip checks are easy: after solving, confirm that surface area equals thirty times square area plus twenty times hexagon area plus twelve times decagon area, and that surface area equals thirty times one plus square root of three plus square root of five plus two square root of five times a squared. If you need only a truncated dodecahedron without the full icosidodecahedron truncation family, use the truncated dodecahedron calculator; if you need a truncated tetrahedron, use that Archimedean sibling; if you need a truncated cone from two radii and height, use the conical frustum calculator instead of forcing an Archimedean model. Keep the edge length inside the published positive range up to one trillion so intermediate cubes and square roots stay finite in IEEE double precision on the Worker and in the browser. Because the formulas only need a, students can recompute by hand for nice radicals and match the API line by line after comparing reported floats to exact expressions involving square roots of three and five. This landing targets people searching for a truncated icosidodecahedron calculator, great rhombicosidodecahedron volume, or Archimedean solid volume and surface area from edge length; it is not a general polyhedron mesh tool, not an icosidodecahedron-only solver, and not a conical frustum calculator. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible measures while learning the contract.

Integration use cases

Homework checks for Archimedean truncated icosidodecahedra

Give students a common edge length; compare their hand-computed volume and surface area against a deterministic API or free browser result, including a negative-edge rejection case.

Read the full KIT documentation →

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