Pentagonal prism calculator
A regular pentagonal prism 5 units tall on a pentagon of side 2 holds 34.409548011779 cubic units, with 50 square units of rectangular walls. Get the volume, wall area and total skin of five-sided columns, labels or insulation wrap. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side length
- 2
- Height
- 5
- Volume
- 34.4095480118
- Lateral surface area
- 50
- Total surface area
- 63.7638192047
- Base area
- 6.8819096024
A right regular pentagonal prism is a solid with two congruent regular pentagon bases joined by five rectangular lateral faces perpendicular to those bases. Given a positive base edge a (the common side length of each pentagon) and a positive height h, three closed-form measures answer most practical questions at once. Volume is the regular-pentagon base area times height—the interior capacity of a uniform five-sided column or packaging blank. Lateral surface area is five times a times h—the wall area only, which matters for labels, insulation wrap, and side coating. Total surface area adds both pentagonal bases to that wall for full paint, material skin, or heat-transfer surface.
What volume and surface area mean for a regular pentagonal prism
A right regular pentagonal prism has two congruent regular pentagon bases whose corresponding vertices are joined by edges perpendicular to both base planes; that perpendicular distance is the height h, and each base edge has length a. The base area of a regular pentagon is one fourth times the square root of five times five plus two times the square root of five, all multiplied by a squared—an exact closed form equivalent to five fourths a squared times cot of pi over five.
Lateral surface area unwraps the five rectangular walls into a single strip of width equal to the perimeter five a and height h, giving L equals five a h. Total surface area adds the two bases: T equals two times base area plus five a h. Units stay consistent with whatever length unit you pass for a and h—metres yield cubic metres of volume and square metres of area; inches yield cubic and square inches.
A useful classroom check is a equals two and h equals five: perimeter is ten, lateral area is fifty, and base area is the closed pentagon formula at side two, so volume and total surface are fixed numbers you can recompute by hand. The arithmetic is elementary multiplications and one square root after coercion, so there is no iteration, no Monte Carlo sampling, no dependence on wall-clock time, and no need for apothem or coordinates of vertices in the input unless you derive them yourself.
Related solids such as irregular pentagon prisms or oblique prisms need different formulas and are out of scope here.
What you can do with it
Packaging and column takeoff
Turn a regular pentagon edge and extrusion height into volume for capacity estimates, or wall and total skin area for wrap, paint, and material orders.
CAD and manufacturing pre-checks
Validate stock pentagonal prism dimensions in scripts before nesting or quoting: volume for mass estimates, lateral and total area for coating and heat treatment.
FAQ
What is the formula for the volume of a regular pentagonal prism?
With base edge a and height h, volume is base area times h, where base area is (1/4)·√(5(5+2√5))·a². Edge and height must use the same length unit.
What is lateral surface area versus total surface area?
Lateral surface area is the five rectangular walls only: 5·a·h.
Why are zero or negative edge and height rejected?
A solid right regular pentagonal prism needs strictly positive edge and height.
Does this handle irregular pentagon bases or oblique prisms?
No. This tool models a right prism with regular (equilateral, equiangular) pentagon bases from outer edge and height only. Irregular footprints and slanted axes need different formulas.
Is the pentagonal prism 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-prism \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2,"height":5}'const res = await fetch("https://api.kit.forhosting.com/geom/pentagonal-prism", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"edge": 2,
"height": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/pentagonal-prism",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"edge": 2,
"height": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/pentagonal-prism", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"edge":2,"height":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"edge":2,"height":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/pentagonal-prism", 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,
"height": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.pentagonal_prism",
"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.
Limits
max_edge | 1000000000000 |
min_edge | 1e-15 |
max_height | 1000000000000 |
min_height | 1e-15 |
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. Non-positive edge or height is rejected with invalid_input so scripts and grading pipelines see an honest error instead of a silent absolute value or a zero-volume solid.
What volume and surface area mean for a regular pentagonal prism
Volume follows at once as base area times height: every cross-section parallel to the bases is the same regular pentagon, so the solid is a straight prism with a pentagonal footprint. Zero or negative dimensions are not accepted under this contract because they do not describe a solid right prism of positive extent; the API returns invalid_input rather than inventing a magnitude.
How to call the API and what comes back
Both must be finite numbers strictly greater than zero and inside the published limits. Use it from geometry homework pipelines, packaging and column takeoff scripts, interview-prep sandboxes, CAD pre-checks, or any workflow that needs reproducible pentagonal prism numbers without installing a computer-algebra package on every machine. Send a JSON body with edge and height. Field edge is the positive side length of each regular pentagon base; aliases a, side, and base_edge are accepted. Field height is the positive prism height; aliases h and H map to the same meaning. On success the response echoes edge and height; returns volume, lateral_surface_area, and total_surface_area as the primary results; and includes base_area for one regular pentagon, perimeter of the base, method set to regular_pentagonal_prism, and a short formula block documenting the closed forms. Failed validation—missing a required field, a non-finite number, a non-positive edge or height, 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 and zero marginal unit price. 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 edge and height in the same unit before you send the request.
Practical tips, limits, and related searches
When teaching, prefer paired examples: edge two and height five for a clean lateral area of fifty; edge one and height one for a unit-side prism whose base area equals the classic regular-pentagon constant; and a zero or negative dimension to demonstrate the hard rejection path. Prefer this tool when you already know the regular base edge and the right prism height and need volume plus both surface measures in one shot. Round-trip checks are easy: after solving, confirm that lateral surface equals perimeter times height, that total surface equals two times base area plus lateral surface, and that volume equals base area times height within floating tolerance. Keep edge and height inside the published positive range up to one trillion so intermediate products stay finite in IEEE double precision on the Worker and in the browser. Because the formulas only need a and h, students can recompute by hand with the closed pentagon area and match the API line by line within floating tolerance. This landing targets people searching for a pentagonal prism calculator or volume and surface area from base edge and height; it is not a cylinder, hexagonal prism, or pyramid tool. 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. If you need a circular right cylinder instead, use the cylinder volume calculator; if you need planar pentagon area without a third dimension, derive base_area alone from the same closed form without calling a height-based prism path.
Integration use cases
Homework checks for pentagonal prism volume and surface area
Give students a base edge and height; compare their hand-computed volume, lateral area, and total surface against a deterministic API or free browser result.