Pentagonal antiprism calculator
A pentagonal antiprism with every edge 1 encloses 1.578689325833 cubic units and has 7.7710818201 square units of surface: two pentagons plus a twisted band of ten equilateral triangles. Enter the edge for models, nets or mesh assets. Free in your browser.
- Free
- Runs in your browser
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- Your data stays on your device
Run — free
Worked example
- Edge
- 1
- Volume
- 1.5786893258
- Surface area
- 7.7710818201
- Bases area
- 3.4409548012
- Lateral area
- 4.3301270189
A uniform pentagonal antiprism is a prismatic polyhedron with two congruent regular pentagon bases, twisted by thirty-six degrees relative to each other, and joined by a band of ten equilateral triangular faces. When every edge shares one positive length a, the solid is completely determined up to congruence: volume equals five plus two times the square root of five, all over six, times a cubed, and total surface area equals five-halves times the cotangent of pi over five plus the square root of three, times a squared. Students meet the figure as the belt of a regular icosahedron after two opposite pentagonal pyramids are removed; makers and CAD users need the same numbers when a brief lists only the edge.
What a uniform pentagonal antiprism is and which metrics the edge determines
In Euclidean solid geometry a uniform n-gonal antiprism has two parallel regular n-gon bases rotated by one hundred eighty degrees over n, connected by two n equilateral triangles when all edges are equal. For n equals five the bases are regular pentagons, the lateral belt has ten equilateral triangles, and the figure is often abbreviated pap in polyhedron catalogues.
Unlike a right regular pentagonal prism, whose lateral faces are five rectangles and whose bases sit without twist, the antiprism twists one base by thirty-six degrees so that each base vertex joins two triangular faces rather than a rectangular wall. With a single positive edge length a shared by every base edge and every lacing edge, two classical closed forms answer the usual homework and design questions. Surface area is (5/2) · (cot(π/5) + √3) · a², the sum of two regular-pentagon areas and ten equilateral-triangle areas.
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. The arithmetic is elementary multiplications and a few square roots after coercion, so there is no iteration, no Monte Carlo sampling, and no dependence on wall-clock time. Related solids such as irregular or star antiprisms, unequal base versus lacing edges, or right pentagonal prisms need different formulas and are out of scope here.
What you can do with it
Polyhedron catalogue and model takeoff
Turn a uniform edge into volume for mass or capacity estimates and total surface for paint, wrap, or material skin when building paper nets or mesh assets of pap.
CAD and manufacturing pre-checks
Validate stock pentagonal-antiprism dimensions in scripts before nesting or quoting: volume for density estimates, surface for coating and heat treatment.
FAQ
What is the formula for the volume of a uniform pentagonal antiprism?
With common edge a, volume is ((5 + 2√5) / 6) · a³. Edge must be positive and finite.
What is the surface area of a uniform pentagonal antiprism?
Surface area is (5/2) · (cot(π/5) + √3) · a²: two regular pentagons plus ten equilateral triangles, all of edge a.
Why are zero or negative edges rejected?
A solid uniform pentagonal antiprism needs a strictly positive edge.
How is this different from a pentagonal prism?
A right pentagonal prism has untwisted bases and five rectangular lateral faces. A pentagonal antiprism twists the bases by 36° and uses ten triangular lateral faces.
Is the pentagonal antiprism 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
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/pentagonal-antiprism \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/pentagonal-antiprism", {
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/pentagonal-antiprism",
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/pentagonal-antiprism", 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/pentagonal-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": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.pentagonal_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
This capability accepts a strictly positive edge, rejects non-positive or non-finite values with invalid_input, and returns volume and surface area as deterministic JSON.
What a uniform pentagonal antiprism is and which metrics the edge determines
Volume is ((5 + 2√5) / 6) · a³, which matches the general uniform antiprism formula at n = 5 and the known unit-edge value about 1.57869. Zero or negative edge is not accepted under this contract because it does not describe a solid of positive extent; the API returns invalid_input rather than inventing a magnitude.
How to call the API and what comes back
Use it from solid-geometry homework pipelines, polyhedron catalogue checks, interview-prep sandboxes, CAD pre-checks, or any workflow that needs reproducible pentagonal-antiprism numbers without installing a computer-algebra package on every machine. Scripts that already call the pentagonal prism calculator for rectangular walls should switch here when the brief specifies twisted bases and triangular lateral faces rather than a straight prism column. Send a JSON body with a single required field named edge. The value is the positive common edge length of the uniform pentagonal antiprism; aliases a, side, and length are accepted. The number must be finite, strictly greater than zero, and inside the published limits so intermediate products stay finite in IEEE double precision on the Worker and in the browser. On success the response echoes edge; returns volume and surface_area as the primary results; includes bases_area for the two pentagons, lateral_area for the ten triangles, method set to uniform_pentagonal_antiprism, and a short formula block documenting the closed forms. Failed validation—missing the required field, a non-finite number, a zero or negative 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 unit before you send the request.
Practical tips, limits, and related searches
Keep the edge inside the published positive range up to one trillion so cubes and products stay finite in double precision. When teaching, prefer paired examples: edge one for the classic unit solid whose volume is (5 + 2√5) / 6 and surface about 7.771; edge two for a scaled solid whose volume scales by eight and surface by four; and a negative or zero edge to demonstrate the hard rejection path. Prefer this tool when you already know the uniform edge and need volume plus total surface in one shot. Because the formulas only need a, students can recompute by hand with the closed volume constant and the face-area sum and match the API line by line within floating tolerance. This landing targets people searching for a pentagonal antiprism calculator or volume and surface area from edge length; it is not a pentagonal prism, hexagonal antiprism, or icosahedron tool, even though a regular icosahedron can be built by attaching two pentagonal pyramids to this belt. 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. Round-trip checks are easy: after solving, confirm that surface_area equals bases_area plus lateral_area, that volume scales as the cube of the edge, and that surface scales as the square of the edge within floating tolerance. If you need a right regular pentagonal prism with rectangular walls, use the pentagonal prism calculator; if you need the regular octahedron as a triangular antiprism, use the octahedron calculator; if you need only planar pentagon area without a third dimension, derive bases_area alone from the regular-pentagon closed form without calling an antiprism path.
Integration use cases
Homework checks for pentagonal antiprism volume and surface
Give students a positive edge length; compare their hand-computed volume and surface area against a deterministic API or free browser result, including rejection of non-positive edges.