Hexagonal antiprism calculator
Enter one edge length and get the volume, total surface and height of a uniform hexagonal antiprism: edge 1 gives a surface of exactly 6√3. Both hexagons and all twelve triangles are 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
- 2.337541789
- Surface area
- 10.3923048454
- Height
- 0.8555996772
A uniform hexagonal antiprism is a convex polyhedron with two parallel regular hexagonal faces, rotated relative to each other by thirty degrees, and twelve equilateral triangular faces that form a zigzag equatorial belt between those bases. Given a single positive edge length shared by every edge of the solid, this calculator returns the enclosed volume, the total surface area of both hexagons and all twelve triangles, and the perpendicular height between the base planes using the classic closed-form formulas of uniform antiprism geometry.
How volume and surface area are computed
Label the common edge length of every face as edge a. For a uniform hexagonal antiprism the two bases are regular hexagons of side a and the twelve lateral faces are equilateral triangles of side a. Each regular hexagon has area (3√3/2) a², so the two bases contribute 3√3 a².
Each equilateral triangle has area (√3/4) a², so twelve triangles contribute 3√3 a². Total surface area is therefore 6√3 a² — a compact closed form that depends only on a. Volume uses the general uniform n-gonal antiprism formula specialized at n equals six: with c equals cos(π/(2n)) and s equals sin(π/n), volume equals n a³ (4c² − 1) to the power three-halves, divided by twenty-four times c times s.
That expression comes from the prismatoid rule V equals (h/3)(B + 2M) where the two bases have area B of a regular n-gon and the midsection M is a regular 2n-gon of circumradius R cos(π/(2n)), together with the height h that makes every lateral edge equal to a. As a sanity check, the same formula at n equals three recovers the regular octahedron volume (√2/3) a³, which is the triangular antiprism case.
Classic checks include edge one, which yields surface area 6√3 approximately 10.392304845413 and a volume near 2.337541788961; edge two scales surface by four and volume by eight. Because the path uses only multiplication, division, cosine, sine, and square roots of fixed trigonometric constants, it is deterministic, O(1), and free of iterative solvers.
Where hexagonal antiprism calculations show up in real work
Secondary-school and college solid-geometry courses ask for the volume and surface area of antiprisms and other uniform polyhedra given only the edge length, often alongside prisms, pyramids, and Platonic solids; a trusted calculator with a strict positive-edge gate speeds homework grading and removes spreadsheet mistakes where someone applies a right-prism formula to a twisted dual-hex solid or forgets the twelve triangular faces.
Architecture, exhibition design, and fabrication workflows size hexagonal lanterns, twisted pavilion nodes, decorative dual-hex shells, and display stands from a single design edge; volume feeds fill, packing, and concrete estimates while total surface area feeds cladding, paint, and sheet cut lists before faces are detailed. Packaging and product design sometimes use antiprism bodies for point-of-sale geometry and dual-hex containers; deterministic metrics support quotes without a full mesh library.
Game, graphics, and CAD libraries store antiprism primitives for colliders, lighting volumes, and procedural props; echoing method plus the input edge makes debug property panels self-describing. If you later need a hexagonal prism without the twist, a hexagonal pyramid, an irregular antiprism with independent height, or only base area without the solid, treat those as separate search intents rather than overloading this hexagonal-antiprism landing.
What you can do with it
Homework and exam checks
Compute uniform hexagonal antiprism volume and surface area from edge length, and reject a negative or zero edge with a clear error.
Lanterns, stands and fabrication
From design edge, recover fill volume and cladding surface for dual-hex shells, twisted nodes and display geometry.
CAD and game antiprism metrics
Turn a single edge into stable volume, height and surface figures for layout, material estimates and property panels.
FAQ
What happens if edge is zero or negative?
Edge must be strictly greater than zero; a negative edge is not a valid antiprism edge length. The error message names the field and the positivity rule.
Is this a uniform (equilateral) antiprism?
Yes. All edges equal the given length: both hexagonal bases and the twelve lateral triangles are regular, and the relative twist between bases is π/6 as required for a uniform hexagonal antiprism.
What does it cost?
The browser tool is free and runs locally.
How is the surface area of a hexagonal antiprism worked out?
The two regular hexagons give 3√3 times the edge squared, and the twelve equilateral triangles give the same again. Together that makes 6√3 times the edge squared.
How do the results change if I double the edge?
The surface area grows four times and the volume eight times, because area follows the square of the edge and volume follows its cube. The height simply doubles.
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/hexagonal-antiprism \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":1}'const res = await fetch("https://api.kit.forhosting.com/geom/hexagonal-antiprism", {
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/hexagonal-antiprism",
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/hexagonal-antiprism", 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/hexagonal-antiprism", 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.hexagonal_antiprism",
"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
Any negative or zero 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
Huge magnitudes can still overflow IEEE-754 double range; when any result is non-finite the request fails with a clear overflow message instead of returning Infinity in JSON.
What you send and what you get back
Provide one required finite number: edge is the common edge length of the uniform hexagonal antiprism and must be strictly greater than zero. There is no optional inradius, circumradius, or independent height on this landing — convert those on the caller side if you start from radii instead of edge. Example checks: edge one recovers surface 6√3 and the n equals six volume formula above; 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 antiprism formulas in every language. Zero and negative values are rejected; a negative edge is never a valid solid-geometry input and produces invalid_input rather than a signed nonsense volume. Missing fields, empty strings, non-numeric text, NaN, and Infinity are rejected with invalid_input and are not charged. On success the JSON response returns edge, volume, surface_area, height (the perpendicular distance between the two hexagonal base planes for the uniform case), and method set to uniform_hexagonal_antiprism_volume_surface so logs and teaching UIs can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where hexagonal antiprism 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?
Surface area = 6√3 a² (two regular hexagons plus twelve equilateral triangles). Volume uses the uniform n-gonal antiprism formula at n=6: V = n a³ (4c²−1)^{3/2} / (24 c sin(π/n)) with c = cos(π/(2n)). The response includes method: "uniform_hexagonal_antiprism_volume_surface".
Does surface_area include the bases?
Yes. surface_area is the total surface area of both hexagonal bases plus all twelve equilateral triangular faces.