Pentagonal bipyramid calculator
A pentagonal bipyramid with edge 1 holds 0.603005664792 cubic units and has 4.330127018922 square units of surface across its ten equilateral faces. Enter one edge to get volume, surface, apex-to-apex height and the middle pentagon. 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
- 0.6030056648
- Surface area
- 4.3301270189
- Height
- 1.0514622242
- Half height
- 0.5257311121
A regular pentagonal bipyramid is the Johnson solid J13 formed by joining two congruent regular pentagonal pyramids base-to-base along a shared equatorial regular pentagon, so that ten equilateral triangular faces meet at two opposite apices on the axis through the pentagon center and every edge has the same length. Given that single positive edge length, this calculator returns the enclosed volume, the total surface area of the ten triangular faces, the apex-to-apex height, the half-height from equator to either apex, and the equatorial pentagon area, using the classic closed-form formulas from solid geometry.
How volume and surface area are computed
Label the common edge length of every face and of the equatorial pentagon as edge a. The regular pentagonal bipyramid (Johnson solid J13) is two congruent regular pentagonal pyramids joined base-to-base so that all ten faces are equilateral triangles. Total apex-to-apex height is twice that half-height.
The area of the equatorial regular pentagon is five-fourths a squared times cotangent of pi over five. Each half is a pyramid of height h on that base, so full volume is two-thirds times equatorial area times h, which closes to volume equals (5/6) times the square root of (3 plus square root of five) over ten, times a cubed. Classic checks include edge one: volume about 0.603006 and surface about 4.330127; edge two scales volume by eight and surface by four.
Because the path uses only multiplication, division, and square roots of five and three, it is deterministic, O(1), and free of iterative solvers.
Where pentagonal bipyramid 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 pentagonal bipyramid given only the edge length, often alongside other Johnson solids and duals of prisms; a trusted calculator with a strict non-negative edge gate speeds homework grading and removes spreadsheet mistakes where someone uses a single-pyramid formula or forgets that the equator is internal and should not be double-counted in surface area.
Crystallography and mineralogy textbooks describe pentagonal bipyramidal crystal habits; converting a design edge into metric volume and face area supports lab demos and teaching models. Architecture, fabrication, and packaging workflows size dual-tapered pentagonal ornaments, pavilion hubs, and display solids from a common edge; volume feeds fill and material estimates while total surface area feeds cladding and paint takeoffs. Game, graphics, and CAD libraries store regular bipyramid primitives for collision proxies and lighting volumes; echoing method plus the input edge makes debug property panels self-describing.
If you later need a single pentagonal pyramid, a pentagonal prism, a hexagonal bipyramid, or a regular icosahedron, treat those as separate search intents rather than overloading this pentagonal-bipyramid landing.
What you can do with it
Homework and exam checks
Compute regular pentagonal bipyramid volume and surface area from edge length, and reject a negative edge with a clear error.
Crystal models and fabrication
From a single design edge, recover fill volume and face area for dual-tapered pentagonal ornaments, demos, and display solids.
CAD and game bipyramid metrics
Turn edge length into stable volume and surface figures for layout, material estimates, and property panels.
FAQ
What formulas does this calculator use?
Volume V = (5/6)√((3+√5)/10) a³; surface area S = (5√3/2) a² (ten equilateral triangular faces). Height and equatorial area are derived from the same regular geometry.
What happens if edge is negative?
The edge must be strictly greater than zero; the error message names the field and states that edge must not be negative. Negative dimensions are never turned into absolute values silently.
Is this the same as a regular icosahedron?
No. A regular icosahedron has twenty triangular faces; a regular pentagonal bipyramid (Johnson J13) has ten. Their volume formulas differ even though both involve √5.
What does it cost?
The browser tool is free and runs locally.
Is the middle pentagon counted in the surface area?
No. Where the two pyramids are joined, the pentagon is inside the solid, so only the ten equilateral triangles count. Adding the pentagon anyway is a common spreadsheet mistake.
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-bipyramid \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":1}'const res = await fetch("https://api.kit.forhosting.com/geom/pentagonal-bipyramid", {
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/pentagonal-bipyramid",
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/pentagonal-bipyramid", 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/pentagonal-bipyramid", 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.pentagonal_bipyramid",
"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 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
The equatorial regular pentagon has circumradius R equals a divided by two times sine of pi over five; the perpendicular distance from that equatorial plane to either apex is half_height h equals the square root of a squared minus R squared, which simplifies to a times the square root of (5 minus square root of five) over ten. For surface area there is no exterior equatorial face left after the two pyramids are joined; only ten congruent equilateral triangles remain, each of area (√3/4) a squared, so surface_area equals (5√3/2) a squared. 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. That closed form matches solid-geometry textbooks and keeps automated graders aligned with hand computation on the same edge.
What you send and what you get back
Provide one required finite number: edge is the common edge length of the regular pentagonal bipyramid and must be strictly greater than zero. There is no optional circumradius, unequal apex heights, or oblique offset—this landing is the regular centered Johnson J13 solid from a single edge only. Example checks: edge one recovers volume (5/6)√((3+√5)/10) and surface (5√3/2); edge two multiplies volume by eight and surface by four; a negative edge fails naming the field and the non-negative rule. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation without reimplementing pentagonal-bipyramid formulas in every language. Negative, zero, missing, empty, non-numeric, NaN, and Infinity values are rejected with invalid_input and are not charged—negative dimensions always fail the gate as the product contract requires. On success the JSON response returns edge, volume, surface_area, height (apex to apex), half_height (equator to either apex), equatorial_area, and method set to regular_pentagonal_bipyramid_volume_surface so logs and teaching UIs can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where pentagonal bipyramid 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
Does surface_area include the equatorial pentagon?
No. After the two pyramids are joined base-to-base, the equator is interior. surface_area is only the ten outer equilateral triangular faces.