ForHosting KIT · Math

Rhombicuboctahedron calculator

Enter the edge length of a rhombicuboctahedron, the solid of 8 triangles and 18 squares, and get its volume and surface area: an edge of 1 gives about 8.71404520791 and 21.464101615138. Free in your browser.

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Worked example

If you enter

Edge: 1

You get

Volume: 8.7140452079
Surface area: 21.4641016151
Triangle area: 0.4330127019
Square area: 1

A rhombicuboctahedron is an Archimedean solid with eight equilateral triangular faces and eighteen square faces, all sharing one common edge length. The uniform case has twenty-four equal edges and twenty-four vertices where three faces meet in a fixed cyclic order. Given a single positive edge length a, closed-form formulas answer the two measures people look up most often: volume equals one third times twelve plus ten times the square root of two, times a cubed; and total surface area equals eighteen plus two times the square root of three, 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 rhombicuboctahedron

A uniform rhombicuboctahedron 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 eight equilateral triangular faces and eighteen square faces arranged so that triangles and squares alternate around the equatorial belts of a cube-like envelope. Surface area is the sum of eight equilateral-triangle areas and eighteen square areas.

Each triangle has area square root of three over four times a squared; each square has area a squared. Multiplying and adding yields the compact closed form eighteen plus two times the square root of three, times a squared. Volume uses the classical Archimedean formula V equals one third times twelve plus ten times the square root of two, times a cubed—the enclosed space of the solid when all twenty-four 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 one third of twelve plus ten square root of two, surface is eighteen plus two square root of three. 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 cube, a cuboctahedron with only fourteen faces, or a truncated cuboctahedron with hexagons and octagons need different formulas and are out of scope here.

What you can do with it

Packing, takeoff, and material estimates

Turn a rhombicuboctahedron 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 rhombicuboctahedron 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 rhombicuboctahedron?

With common edge length a, volume is (1/3)·(12 + 10√2)·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 rhombicuboctahedron?

Total surface area is (18 + 2√3)·a²: eighteen squares each of area a² plus eight equilateral triangles each of area (√3/4)·a².

How is a rhombicuboctahedron different from a cuboctahedron?

A cuboctahedron has eight triangles and six squares (fourteen faces). A rhombicuboctahedron has eight triangles and eighteen squares (twenty-six faces) and a different vertex figure.

Why is a negative edge rejected?

A solid rhombicuboctahedron needs a strictly positive edge.

Is the rhombicuboctahedron 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/rhombicuboctahedron

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/rhombicuboctahedron \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"edge":1}'

const res = await fetch("https://api.kit.forhosting.com/geom/rhombicuboctahedron", {
  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/rhombicuboctahedron",
    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/rhombicuboctahedron", 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/rhombicuboctahedron", 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.rhombicuboctahedron",
  "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 rhombicuboctahedron 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 a design series. Send a JSON body with a single required field edge—the positive common edge length of the uniform rhombicuboctahedron. 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 triangle_area for one equilateral face, square_area for one square face, method set to rhombicuboctahedron, 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 required by the product contract. 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 eight times triangle area plus eighteen times square area, and that surface area equals eighteen plus two square root of three times a squared. If you need only a cube, use the cube calculator; if you need a regular octahedron or icosahedron, use those dedicated edge-length solvers instead of forcing a rhombicuboctahedron 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 two and three. This landing targets people searching for a rhombicuboctahedron calculator or Archimedean solid volume and surface area from edge length; it is not a general polyhedron mesh tool, not a cuboctahedron-only solver, and not a truncated-cuboctahedron 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 rhombicuboctahedra

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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