Decagonal prism calculator
Enter the base side and height of a regular decagonal prism and get its capacity and surfaces: side 2 and height 5 give a base perimeter of 20 and a wall area of 100, plus volume and full surface. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side
- 2
- Height
- 5
- Volume
- 153.8841768588
- Lateral surface area
- 100
- Total surface area
- 161.5536707435
- Base area
- 30.7768353718
A right regular decagonal prism is a solid with two congruent regular ten-sided bases joined by ten rectangular lateral faces perpendicular to those bases. Given a positive base side a (the common edge length of each regular decagon) and a positive height h, three closed-form measures answer most practical questions at once. Volume is the regular-decagon base area times height—the interior capacity of a uniform ten-sided column, packaging blank, or architectural extrusion. Lateral surface area is ten times a times h—the wall area only, which matters for labels, insulation wrap, and side coating. Total surface area adds both decagonal bases to that wall for full paint, material skin, or heat-transfer surface.
What volume and surface area mean for a regular decagonal prism
A right regular decagonal prism has two congruent regular decagon bases whose corresponding vertices are joined by edges perpendicular to both base planes; that perpendicular distance is the height h, and each base side has length a.
The base area of a regular decagon is five halves times the square root of five plus two times the square root of five, all multiplied by a squared—an exact closed form equivalent to ten fourths a squared times cot of pi over ten, or five halves a squared over tan of pi over ten. Lateral surface area unwraps the ten rectangular walls into a single strip of width equal to the perimeter ten a and height h, giving L equals ten a h.
Total surface area adds the two bases: T equals two times base area plus ten 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 twenty, lateral area is one hundred, and base area is the closed decagon 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 decagon prisms or oblique prisms need different formulas and are out of scope here.
What you can do with it
Packaging and ten-sided column takeoff
Turn a regular decagon side 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 decagonal 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 decagonal prism?
With base side a and height h, volume is base area times h, where base area is (5/2)·√(5+2√5)·a². Side and height must use the same length unit.
What is lateral surface area versus total surface area?
Lateral surface area is the ten rectangular walls only: 10·a·h.
Why are negative side and height rejected?
A solid right regular decagonal prism needs strictly positive side and height.
Does this handle irregular decagon bases or oblique prisms?
No. This tool models a right prism with regular (equilateral, equiangular) decagon bases from outer side and height only. Irregular footprints and slanted axes need different formulas.
Is the decagonal 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/decagonal-prism \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side":2,"height":5}'const res = await fetch("https://api.kit.forhosting.com/geom/decagonal-prism", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side": 2,
"height": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/decagonal-prism",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side": 2,
"height": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/decagonal-prism", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"side":2,"height":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side":2,"height":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/decagonal-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
{
"side": 2,
"height": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.decagonal_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.
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. Negative or non-positive side 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 decagonal prism
Volume follows at once as base area times height: every cross-section parallel to the bases is the same regular decagon, so the solid is a straight prism with a ten-sided 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 decagonal prism numbers without installing a computer-algebra package on every machine. When you already know only the apothem of a planar decagon and not the outer side, convert to side first with the regular-polygon side formula, then call this prism path with a consistent height. Send a JSON body with side and height. Field side is the positive edge length of each regular decagon base; aliases a, edge, base_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 side and height; returns volume, lateral_surface_area, and total_surface_area as the primary results; and includes base_area for one regular decagon, perimeter of the base, method set to regular_decagonal_prism, and a short formula block documenting the closed forms. Failed validation—missing a required field, a non-finite number, a negative or non-positive side 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 side and height in the same unit before you send the request.
Practical tips, limits, and related searches
When teaching, prefer paired examples: side two and height five for a clean lateral area of one hundred; side one and height one for a unit-side prism whose base area equals the classic regular-decagon constant; and a negative dimension to demonstrate the hard rejection path required by the product contract. Prefer this tool when you already know the regular base side 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 side 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 decagon area and match the API line by line within floating tolerance. This landing targets people searching for a decagonal prism calculator or volume and surface area from base side and height; it is not a cylinder, hexagonal prism, pentagonal 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 decagon area without a third dimension, use a regular-decagon area tool or derive base_area alone from the same closed form without calling a height-based prism path.
Integration use cases
Homework checks for decagonal prism volume and surface area
Give students a base side and height; compare their hand-computed volume, lateral area, and total surface against a deterministic API or free browser result.