ForHosting KIT · Math

Decagon circumradius calculator

Get the radius of the circle through all ten corners of a regular decagon from its side length, so a side of 2 gives a circumradius of about 3.2360679775. The radius is always the side times phi (φ). Free in your browser.

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Worked example

If you enter

Side: 2

You get

Circumradius: 3.2360679775

The circumradius of a regular decagon is the radius of the unique circle that passes through all ten vertices—the distance from the centre to any corner. Architects, coin and medal designers, game developers, and CNC fabricators often know the edge length of a ten-sided plate, tile, or map cell and need that outer radius for bounding circles, clearance, packing, or layout.

What the circumradius of a regular decagon is and why the formula is closed form

A regular decagon has ten equal sides and ten equal interior angles of one hundred forty-four degrees. Connecting the centre to every vertex partitions the figure into ten congruent isosceles triangles, each with two radii equal to the circumradius R and a base equal to the side length s. The central angle of each triangle is thirty-six degrees, so half of that is eighteen degrees.

In the right triangle formed by dropping a perpendicular from the centre to a side, the half-side s over two sits opposite the eighteen-degree half-angle and the hypotenuse is R, which yields the general regular n-gon identity R equals s divided by two times the sine of pi over n. With n equal to ten, sine of pi over ten is exactly eighteen degrees, and that sine evaluates in closed form to the square root of five minus one, all over four.

Equivalent checks confirm the identity: R equals s over two sin of thirty-six degrees for the full central angle split, or R equals s times phi. All expressions are algebraically identical for a perfect regular decagon, and the implementation uses the radical form for stable floating-point results without calling trigonometric functions at runtime.

Where the decagon's outer radius helps

Exam-prep generators need a deterministic oracle that marks zero and negative sides as invalid instead of silently returning nonsense radii a buggy grader might accept. Game and map engines that use regular decagon cells for terrain or UI badges convert side length to vertex radius when placing sprites on a circumcircle. Failed validation does not consume balance.

Keep requests small, pin the rounded circumradius in tests, and you get reproducible radii for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.

What you can do with it

CAD, medals, and fabrication clearances

Convert a known edge of a ten-sided plate or medal into the circumscribed-circle radius for packing, laser paths, and outer clearances.

Games and map cell layout

Map regular-decagon tile side lengths to vertex radii when placing assets on a shared circumcircle in engines and level tools.

FAQ

What happens if the side is zero or negative?

A geometric side length must be strictly positive; zero and negative values are rejected and not billed.

Is circumradius the same as apothem?

No. Circumradius reaches the vertices; apothem (inradius) reaches the midpoints of the sides and is smaller. This tool returns only the circumradius.

Why does the number phi appear in the decagon circumradius?

For a regular decagon, the radius to each corner equals the side multiplied by phi, (1 + √5)/2, about 1.618. So the circumradius is always that same multiple of the side.

What is the circumradius of a decagon with side 2?

About 3.2360679775, which is 2 times phi. Use it as the radius of the circle that encloses a ten-sided plate, medal or tile.

Which unit is the result in?

The same unit as the side. Enter the side in millimetres and the circumradius comes out in millimetres; nothing is converted along the way.

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

POSThttps://api.kit.forhosting.com/geom/decagon-circumradius

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-circumradius \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"side":2}'

const res = await fetch("https://api.kit.forhosting.com/geom/decagon-circumradius", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "side": 2
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/geom/decagon-circumradius",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "side": 2
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/decagon-circumradius", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"side":2}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"side":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/decagon-circumradius", 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
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "geom.decagon_circumradius",
  "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

Per request$0.002

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

Zero or negative side lengths fail with invalid_input instead of a silent NaN.

Input contract, response shape, and error behaviour

On success the body includes the echoed side, the circumradius rounded to twelve decimal places for stable goldens across runtimes, and a formula string restating R equals s times one plus the square root of five over two. Keep units consistent yourself: the calculator is unit-agnostic, so if side is in millimetres the circumradius is in millimetres. POST a JSON body with one required field: side, the finite positive side length of the regular decagon. The value may arrive as a JSON number or as a numeric string; non-finite values, missing fields, empty strings, and non-numeric text fail with invalid_input naming the field. Pair the response with your own drawing code if you need a sketch of the circumscribed circle or the ten radii; this endpoint computes the scalar radius only and does not emit SVG, step-by-step prose beyond the formula field, or related metrics such as area or apothem. Extremely large finite sides remain valid as long as the IEEE floating result stays finite; non-finite intermediate results would fail the finite-number checks on input rather than return Infinity as a success payload.

Where a decagon circumradius API earns its keep

Architecture and product-design notebooks often size ten-sided plates, medals, or hexagonal-adjacent tiling variants from a known edge and need the bounding circle for packing or laser cutouts; calling a tiny parse API avoids pulling a full computer-algebra system for a one-parameter special case.

Integration use cases

Classroom and homework checks

Let students submit a positive side length and verify the regular-decagon circumradius against the golden-ratio formula R=s·(1+√5)/2.

Technical questions

How do I compute the circumradius of a regular decagon from side length?

Send a finite positive number side. The API returns R = s·(1+√5)/2, the radius of the circle through all ten vertices, plus a short formula string.

Why does the formula involve the golden ratio?

For a regular decagon, sin(π/10)=(√5−1)/4, so R=s/(2·sin(π/10)) simplifies exactly to s·(1+√5)/2, which is s times the golden ratio φ.

What does the decagon circumradius calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

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