Decagon area calculator
Enter one side of a regular ten-sided figure and get its area: a side of 5 gives about 192.355, with a perimeter of 50 plus the apothem and outer radius for the rest of the worksheet. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side
- 5
- Area
- 192.3552210735
- Perimeter
- 50
- Apothem
- 7.6942088429
- Circumradius
- 8.0901699437
A regular decagon has ten equal sides and ten equal interior angles. Given only the side length, this calculator returns the enclosed area using the closed form A equals ten over four times side squared times the cotangent of pi over ten, which simplifies to five halves times the square root of five plus two square roots of five times side squared. Related perimeter, apothem, and circumradius ship in the same response so one request covers the usual worksheet pack.
How regular decagon area is computed from the side
Label the common side length of the regular decagon as side (or s). A regular n-gon with n equal to ten decomposes into ten congruent isosceles triangles that meet at the center; each triangle has two radii and base equal to the side.
The standard closed form is area equals n over four times side squared times the cotangent of pi over n, which for a decagon becomes area equals ten over four times side squared times cot of pi over ten, or equivalently five halves times side squared times cot of pi over ten. Because cot of eighteen degrees equals the square root of five plus two square roots of five, the same area is exactly five halves times that radical times side squared.
The apothem (inradius) equals side over two times tan of pi over ten, the circumradius equals side over two times sin of pi over ten, and the perimeter is simply ten times side. Area also equals one half times perimeter times apothem—the same numeric result from elementary geometry. Classic checks include side one, which yields the unit regular decagon area near seven point six nine four, and side five, which scales by twenty-five to about one hundred ninety-two point three five five.
Because the path uses only multiplication, division, and standard trigonometric functions of a fixed angle—or the exact radical form (5/2)√(5+2√5)s²—it is deterministic, O(1), and free of iterative solvers.
Where decagon area calculations show up in real work
Secondary-school and college geometry courses ask for the area of a regular decagon given only the side, and often for perimeter, apothem, or circumradius on the same problem set; a trusted calculator with a strict positive-side gate speeds homework grading and removes spreadsheet mistakes where someone multiplies side by ten and stops without the cot factor.
Architecture, tiling, and manufacturing workflows size ten-sided floor tiles, decorative medallions, coin-like badges, and regular ten-sided packing layouts from a single edge length; area feeds material estimates while apothem and circumradius help place centerlines and clearance circles. Game, graphics, and CAD libraries occasionally store regular decagonal cells for maps, strategy boards, and procedural meshes; deterministic area and related metrics support collision proxies and layout math without a full computational-geometry stack.
Survey and sketch tools sometimes approximate flat regular panels as decagons when only the side is measured; having a shared numeric oracle keeps field notes and office scripts aligned. If you later need irregular decagon area from vertices, a general regular n-gon with arbitrary side count, or only the perimeter without area, treat those as separate search intents rather than overloading this decagon-area page.
What you can do with it
Homework and exam checks
Compute regular decagon area from side length, and reject zero or negative sides with a clear error.
Tiles, panels, and ten-sided faces
From a measured edge length, recover area, perimeter, and apothem for cut lists and material estimates.
FAQ
What formula does this calculator use?
area = (10/4) × side² × cot(π/10) = (5/2)√(5+2√5) × side² for a regular decagon. Equivalently area = (10 × side²) / (4 × tan(π/10)). Perimeter is 10×side; apothem is side/(2×tan(π/10)).
What happens if the side length is zero or negative?
The side must be a positive finite number; the error message names the field.
Does this work for irregular decagons?
No. This tool assumes a regular decagon (all sides equal, all interior angles equal). Irregular polygons need vertex coordinates and a different method such as the shoelace formula.
What else is returned besides area?
Perimeter (10×side), apothem (inradius), circumradius, sides count (10), the area formula string, and a method tag for logs and teaching UIs.
What does it cost?
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/decagon-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side":5}'const res = await fetch("https://api.kit.forhosting.com/geom/decagon-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/decagon-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/decagon-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"side":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/decagon-area", 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": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.decagon_area",
"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
A missing, non-finite, zero, or negative side is rejected with a clear invalid_input error instead of a silent nonsense area. No network calls, no random sampling, and no floating-point surprises beyond ordinary IEEE-754 arithmetic on the numbers you provide.
How regular decagon area is computed from the side
Huge magnitudes can still overflow IEEE-754 double range; when intermediates are non-finite the request fails with a clear overflow message instead of returning Infinity in JSON. That closed form matches analytic geometry textbooks and keeps automated graders aligned with hand computation on the same side length. The response also echoes perimeter, apothem, circumradius, sides count ten, formula_area, and method so a single call covers the metrics most worksheets ask for next to area.
What you send and what you get back
Provide one required finite number: side is the length of any edge of the regular decagon and must be strictly positive. Example checks: side five yields area about one hundred ninety-two point three five five with perimeter fifty; side one yields the unit regular decagon area near seven point six nine four; a side of zero or minus two fails naming the field. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation without reimplementing the ten-over-four cot formula in every language. Optional precision is an integer from zero through fifteen decimal places (default twelve) for numeric fields in the response; aliases decimals and dp map to the same control. Missing fields, empty strings, non-numeric text, NaN, Infinity, zero, and any negative side are rejected with invalid_input and are not charged. On success the JSON response returns side, area, perimeter, apothem, circumradius, sides set to ten, formula_area describing the closed form including the (5/2)√(5+2√5) simplification, and method set to regular_decagon_side_squared so logs and teaching UIs can document the algorithm. API pricing is the published base of $0.002 per successful item. Failed validation never consumes prepaid balance, which keeps exploratory scripts cheap when learners probe the positive-side gate.
Where decagon area 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.
Integration use cases
Game grids and CAD ten-sided cells
Turn side length into area, apothem, and circumradius for layout and collision proxies in browser or API.