Icosidodecahedron calculator
An icosidodecahedron with an edge of 1 has a surface of about 29.306 square units and a volume of (45 + 17√5)/6 cubic units. Enter one edge and get both, with its faces, edges and vertices counted. 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
- 13.8355259362
- Surface area
- 29.3059828449
- Faces
- 32
- Edges
- 90
An icosidodecahedron is the Archimedean solid obtained by rectifying a regular dodecahedron or a regular icosahedron until every original face becomes a regular polygon meeting alternating triangles and pentagons at each vertex. Students, 3D designers, and exam writers need its volume and total surface area from a single shared edge length when sizing quasi-spherical shells, checking textbook Archimedean tables, or scripting parametric models. This icosidodecahedron calculator accepts one edge length, applies the closed formulas V equals (45 plus 17 times the square root of five) over six times a cubed and S equals (5 times the square root of three plus 3 times the square root of twenty-five plus ten times the square root of five) times a squared, and returns both metrics with the solid’s fixed face, edge, and vertex counts.
What an icosidodecahedron is and which formulas fix volume and surface
Start from a regular dodecahedron or icosahedron and rectify (cut every edge at its midpoint and expand until original faces shrink to regular polygons of the dual type). The resulting convex polyhedron is the icosidodecahedron, one of the thirteen Archimedean solids: all faces are regular polygons, and the vertex figure is the same at every vertex. It has thirty-two faces in total—twenty equilateral triangles and twelve regular pentagons—ninety equal edges, and sixty identical vertices.
Because every edge has the same length a after a uniform rectification, 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 (45 plus 17 times the square root of five) divided by six, times a cubed, and total surface area S equals five times the square root of three plus three times the square root of twenty-five plus ten times the square root of five, times a squared.
Those expressions come from summing the areas of twenty equilateral triangles of side a and twelve regular pentagons of side a, plus the known volume of the fully rectified dodecahedron–icosahedron dual pair 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.
Who uses an icosidodecahedron calculator
Classrooms and tutoring platforms use an icosidodecahedron calculator so learners can check volume and surface area for arbitrary non-negative edge lengths without mixing the triangle and pentagon 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 dodecahedron or icosahedron calculator, this landing targets the narrower search intent of the icosidodecahedron 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 icosidodecahedron into a search box and need volume plus surface area from edge length.
What you can do with it
Classroom checks of icosidodecahedron metrics
Let students submit an edge length and verify volume V=(45+17√5)/6·a³ and surface S=(5√3+3√(25+10√5))·a² against textbook Archimedean solid tables.
CAD and parametric design notebooks
Compute volume and surface area for icosidodecahedron cells in scripts that size decorative polyhedra, quasi-spherical 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
Which formulas does the calculator use?
Volume V = (45 + 17√5)/6 · a³. Surface area S = (5√3 + 3√(25+10√5)) · a², where a is the common edge length of the icosidodecahedron.
What happens if the edge is negative?
An icosidodecahedron requires a non-negative edge length; negative values are rejected and those calls are not charged.
Is the icosidodecahedron calculator free?
The browser tool runs free on your device.
How does an icosidodecahedron differ from a dodecahedron or icosahedron?
A regular dodecahedron has twelve pentagons; a regular icosahedron has twenty triangles. An icosidodecahedron has both: twenty equilateral triangles and twelve regular pentagons (32 faces, 90 edges, 60 vertices) after rectifying either parent solid.
How many faces, edges and vertices does an icosidodecahedron have?
It has 32 faces, 20 equilateral triangles and 12 regular pentagons, plus 90 equal edges and 60 identical vertices where triangles and pentagons alternate. The result lists these counts next to the volume and surface.
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/icosidodecahedron \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/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/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/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/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.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
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 an icosidodecahedron 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 thirteen point eight three six and surface area approximately twenty-nine point three zero six 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 icosidodecahedron. 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-two, edges set to ninety, vertices set to sixty, method set to icosidodecahedron, and two formula strings for display next to the answer: V = (45 + 17√5)/6 · a³ and S = (5√3 + 3√(25+10√5)) · a². Pair the response with your own renderer if you need a 3D mesh of the icosidodecahedron; this endpoint returns scalars and topology only and does not emit meshes, unit conversions, or midradius or circumradius.
Where an icosidodecahedron calculator API earns its keep
CAD, digital-fabrication, and parametric-design notebooks often size Archimedean cells, geodesic-related shells, or decorative polyhedra; calling a tiny parse API avoids pulling a computer-algebra system for two closed multiplications.
Technical questions
How do I compute icosidodecahedron 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.