Dodecagon area calculator
Give the side of a regular dodecagon and get the area it encloses: a side of 1 covers about 11.196 square units, three times (2 + √3) the side squared, with perimeter, apothem and circumradius added. 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
- 279.9038105677
- Perimeter
- 60
- Apothem
- 9.3301270189
- Circumradius
- 9.6592582629
A regular dodecagon has twelve equal sides and twelve equal interior angles. Given only the side length, this calculator returns the enclosed area using the closed form A equals twelve over four times side squared times the cotangent of pi over twelve, which simplifies to three times two plus the square root of three times side squared. Related perimeter, apothem, and circumradius ship in the same response so one request covers the usual worksheet pack.
How regular dodecagon area is computed from the side
Label the common side length of the regular dodecagon as side (or s). A regular n-gon with n equal to twelve decomposes into twelve 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 dodecagon becomes area equals twelve over four times side squared times cot of pi over twelve, or equivalently three times side squared times cot of pi over twelve. Because cot of fifteen degrees equals two plus the square root of three, the same area is exactly three times the quantity two plus square root of three times side squared.
The apothem (inradius) equals side over two times tan of pi over twelve, the circumradius equals side over two times sin of pi over twelve, and the perimeter is simply twelve 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 dodecagon area near eleven point one nine six, and side five, which scales by twenty-five to about two hundred seventy-nine point nine zero.
Because the path uses only multiplication, division, and standard trigonometric functions of a fixed angle—or the exact radical form 3(2+√3)s²—it is deterministic, O(1), and free of iterative solvers.
Where dodecagon area calculations show up in real work
Secondary-school and college geometry courses ask for the area of a regular dodecagon 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 twelve and stops without the cot factor.
Architecture, tiling, and manufacturing workflows size twelve-sided floor tiles, clock faces, decorative rosettes, coin-like medallions, and regular twelve-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 dodecagonal 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 dodecagons when only the side is measured; having a shared numeric oracle keeps field notes and office scripts aligned. If you later need irregular dodecagon area from vertices, a general regular n-gon with arbitrary side count, or a three-dimensional dodecahedron volume, treat those as separate search intents rather than overloading this dodecagon-area page.
What you can do with it
Homework and exam checks
Compute regular dodecagon area from side length, and reject zero or negative sides with a clear error.
Tiles, panels, and twelve-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 = (12/4) × side² × cot(π/12) = 3(2+√3) × side² for a regular dodecagon. Equivalently area = (12 × side²) / (4 × tan(π/12)). Perimeter is 12×side; apothem is side/(2×tan(π/12)).
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 dodecagons?
No. This tool assumes a regular dodecagon (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 (12×side), apothem (inradius), circumradius, sides count (12), 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/dodecagon-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side":5}'const res = await fetch("https://api.kit.forhosting.com/geom/dodecagon-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/dodecagon-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/dodecagon-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/dodecagon-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.dodecagon_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 dodecagon 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 twelve, 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 dodecagon and must be strictly positive. Example checks: side five yields area about two hundred seventy-nine point nine zero with perimeter sixty; side one yields the unit regular dodecagon area near eleven point one nine six; 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 twelve-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 twelve, formula_area describing the closed form including the 3(2+√3) simplification, and method set to regular_dodecagon_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 dodecagon 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 twelve-sided cells
Turn side length into area, apothem, and circumradius for layout and collision proxies in browser or API.