Octagon circumradius calculator
The circle through all eight corners of a regular octagon with sides of 4 has a radius of about 5.22625. Enter the side length to get this circumradius for plates, stop-sign blanks and clearance circles, along with the apothem, perimeter and area. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side
- 4
- Circumradius
- 5.2262518595
- Apothem
- 4.8284271247
- Perimeter
- 32
- Area
- 77.2548339959
A regular octagon has eight equal sides and eight equal interior angles. Given only the side length, this calculator returns the circumradius—the radius of the unique circle that passes through every vertex—using the closed regular-polygon identity R equals side divided by two times the sine of pi over eight. Related apothem, perimeter, and area ship in the same response so one request covers the usual worksheet pack that asks for both the outer circle and the enclosed region.
How regular octagon circumradius is computed from the side
Label the common side length of the regular octagon as side (or s). Every regular n-gon sits on a circle whose center is the common meeting point of the perpendicular bisectors of the sides. Connecting that center to two adjacent vertices forms an isosceles triangle with apex angle two-pi over n and base equal to the side.
Bisecting the apex produces a right triangle with opposite side s over two and hypotenuse equal to the circumradius R, so sin of pi over n equals s over two R, which rearranges to the textbook formula R equals s divided by two times sin of pi over n. For a regular octagon, n is eight, so R equals side divided by two times sine of pi over eight.
Equivalent closed forms without calling sine include R equals side times the square root of one plus the square root of two over two, and R equals half the side times the square root of four plus two times the square root of two; all three agree in floating point for ordinary side lengths. Classic checks include side one, which yields R about 1.30656, and side two, which scales linearly to about 2.61313.
Because the path uses only multiplication, division, and a single sine of a fixed angle, it is deterministic, O(1), and free of iterative solvers. The response also echoes apothem (distance from center to a side midpoint), perimeter eight times side, and area one half times perimeter times apothem so a single call covers the metrics most worksheets ask for next to the circumradius.
Where octagon circumradius calculations show up in real work
Secondary-school and college geometry courses constantly ask for the radius of the circle through the vertices of a regular octagon given only the side, and often for apothem or area on the same problem set; a trusted calculator with a strict positive-side gate speeds homework grading and removes spreadsheet mistakes where someone confuses inradius with circumradius.
Architecture, signage, and manufacturing workflows size octagonal plates, stop-sign blanks, decorative tiles, and clearance circles around regular eight-sided panels from a single edge length; circumradius feeds nesting and hole-circle layouts while apothem and area help place centerlines and material estimates. Game, graphics, and CAD libraries store regular octagonal cells for maps, strategy boards, and procedural meshes; deterministic outer radius 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 octagons when only the side is measured; having a shared numeric oracle keeps field notes and office scripts aligned. If you later need irregular octagon metrics from vertices, a general regular n-gon solver, or octahedron solid geometry, treat those as separate search intents rather than overloading this octagon-circumradius landing.
What you can do with it
Homework and exam checks
Compute regular octagon circumradius from side length, and reject zero or negative sides with a clear error.
Plates, signs, and hole circles
From a measured edge length, recover outer circle radius, apothem, and area for cut lists and bolt patterns.
FAQ
What formula does this calculator use?
R = side / (2 × sin(π/8)) for a regular octagon. Equivalent closed forms use √(1 + √2/2).
What happens if the side length is zero or negative?
The side must be a positive finite number; the error message names the field and requires a positive value.
Is circumradius the same as apothem?
No. Circumradius is the distance from center to a vertex (outer circle). Apothem is the distance from center to a side midpoint (inner circle). Both are returned so you can compare them.
What else is returned besides circumradius?
Apothem, perimeter (8×side), area (½×perimeter×apothem), sides (8), the circumradius 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/octagon-circumradius \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side":4}'const res = await fetch("https://api.kit.forhosting.com/geom/octagon-circumradius", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/octagon-circumradius",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/octagon-circumradius", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"side":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/octagon-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": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.octagon_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
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 radius. No network calls, no random sampling, and no floating-point surprises beyond ordinary IEEE-754 arithmetic on the numbers you provide.
How regular octagon circumradius 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.
What you send and what you get back
Provide one required finite number: side is the length of any edge of the regular octagon and must be strictly positive. Example checks: side four yields circumradius about 5.22625 with perimeter thirty-two; side one yields the unit regular octagon outer radius; a side of zero or minus two fails naming the field and requiring a positive value. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation without reimplementing the two-sine 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, circumradius, apothem, perimeter, area, sides set to eight, formula_circumradius describing the closed form, and method set to regular_octagon_side_to_circumradius so logs and teaching UIs can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where octagon circumradius 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
Octagon cells in games and CAD
Turn side length into circumradius and perimeter for layout, packing, and collision proxies in browser or API.