Icosahedron calculator
Type one edge length and get the volume and the surface of all 20 triangular faces: an edge of 1 gives a volume of (5/12)(3+√5) and a surface of 5√3. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Edge
- 2
- Volume
- 17.453559925
- Surface area
- 34.6410161514
A regular icosahedron is one of the five Platonic solids: twenty congruent equilateral triangular faces, thirty edges of equal length, and twelve vertices where five triangles meet. Given a single positive edge length, this calculator returns the enclosed volume and the total surface area of all faces.
How volume and surface area are computed
Label the common edge length of every face as a. The volume of a regular icosahedron is five-twelfths times three plus the square root of five, times a cubed: volume equals (5/12)(3+√5) a³. The total surface area is twenty times the area of one equilateral triangle of side a.
Classic checks include edge one, which yields volume about 2.181694990625 and surface about 8.660254037844; edge two scales volume by eight and surface by four, giving about 17.453559925 and 34.641016151378; edge five is a common larger case with volume about 272.711873828114 and surface about 216.50635094611. Because the path uses only multiplication, a fixed rational coefficient, Math.sqrt(5), and Math.sqrt(3), it is deterministic, O(1), and free of iterative solvers.
Where icosahedron calculations show up in real work
Secondary-school and college solid-geometry courses constantly ask students for the volume and surface area of a regular icosahedron given only the edge length, often alongside cube, tetrahedron, octahedron, and dodecahedron problems; a trusted calculator with a strict positive-edge gate speeds homework grading and removes spreadsheet mistakes where someone confuses face count or uses the wrong radical.
Architecture and fabrication workflows size geodesic dome nodes, pavilion panels, and decorative platonic shells from a design edge; volume feeds material fill and packing estimates while surface area informs cladding, paint, and sheet-metal cut lists. Tabletop games, dice manufacturers, and prop designers work with twenty-sided dice (d20) whose body is a regular icosahedron; edge-to-volume and edge-to-surface metrics support mold design and material estimates before CAD.
Game, graphics, and CAD libraries store icosahedral colliders and subdivision bases for spheres; deterministic volume and surface metrics support debug readouts and property panels without a full mesh library. Chemistry and molecular visualization sometimes use icosahedral cages and duals of dodecahedral carbon frameworks; the same pure formulas give consistent scale checks. If you later need a truncated icosahedron (buckyball), an irregular polyhedron, or radii instead of edge as the primary input, treat those as separate search intents rather than overloading this regular-icosahedron landing.
What you can do with it
Homework and exam checks
Compute regular icosahedron volume and total surface area from edge length, and reject non-positive edge with a clear error.
Geodesic nodes, dice and fabrication
From design edge, recover capacity volume and cladding surface for shells, molds and cut lists.
CAD and game icosahedron metrics
Turn a single edge into volume and surface figures for layout, material estimates and property panels.
FAQ
What happens if the edge is zero or negative?
The edge must be positive (strictly greater than zero); the error message names the field.
Is this a truncated icosahedron or buckyball?
No. This capability is the regular Platonic icosahedron only—twenty equilateral triangles. Truncated forms (soccer ball / fullerene shells) use different face counts and formulas.
Can I pass inradius or circumradius instead of edge?
Not on this landing. Convert radius to edge on the caller side, then send edge.
What does it cost?
The browser tool is free and runs locally.
How much do volume and surface grow if I double the edge?
Volume follows the cube of the edge and surface follows its square, so an edge of 2 gives eight times the volume and four times the surface of an edge of 1. The same rule lets you rescale a model without recomputing from scratch.
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/icosahedron \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/icosahedron", {
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/icosahedron",
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/icosahedron", 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/icosahedron", 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.icosahedron",
"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
A non-positive edge is rejected with a clear invalid_input error instead of a silent nonsense volume. No network calls, no random sampling, and no floating-point surprises beyond ordinary IEEE-754 arithmetic on the edge you provide.
How volume and surface area are computed
Each triangle has area (√3/4) a², so surface_area equals 5 √3 a². Huge magnitudes can still overflow IEEE-754 double range; when intermediates are non-finite the request fails with a clear overflow message instead of returning Infinity in JSON. That closed form keeps automated graders aligned with hand computation on the same edge length.
What you send and what you get back
Provide one required finite number: edge is the edge length of the regular icosahedron and must be strictly greater than zero. Zero and negative values are rejected; there is no optional inradius or circumradius on this landing—convert those to edge on the caller side if needed. Example checks: edge one yields volume (5/12)(3+√5) and surface 5√3; edge two multiplies volume by eight and surface by four; a zero or negative edge fails naming the field and the positivity rule. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation without reimplementing the Platonic formulas in every language. Missing fields, empty strings, non-numeric text, NaN, Infinity, and any non-positive edge are rejected with invalid_input and are not charged. On success the JSON response returns edge, volume, surface_area, and method set to regular_icosahedron_volume_surface so logs and teaching UIs can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where icosahedron calculations show up in real work
Because this capability is a pure parse engine, failed validation does not consume balance and successful runs stay at $0.002 per item.
Technical questions
What formulas does this calculator use?
volume = (5/12)(3+√5) a³; surface_area = 5√3 a² (twenty equilateral triangles). The response includes method: "regular_icosahedron_volume_surface".