Nonagon apothem calculator
A regular nonagon with sides of 4 has an apothem of about 5.49495, the distance from its center to the middle of each side. Enter the side length and get the apothem along with the circumradius, 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
- Apothem
- 5.4949548389
- Circumradius
- 5.8476088003
- Perimeter
- 36
- Area
- 98.9091871004
A regular nonagon has nine equal sides and nine equal interior angles. Given only the side length, this calculator returns the apothem—the perpendicular distance from the center to the middle of any side, also called the inradius—using the closed regular-polygon identity apothem equals side divided by two times the tangent of pi over nine. Related circumradius, perimeter, and area ship in the same response so one request covers the usual worksheet pack that asks for both the inner offset and the enclosed region.
How regular nonagon apothem is computed from the side
Label the common side length of the regular nonagon as side (or s). Every regular n-gon has an inscribed circle that touches each side at its midpoint; the radius of that circle is the apothem. Connecting the 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 adjacent leg equal to the apothem a, so tan of pi over n equals s over two a, which rearranges to the textbook formula a equals s divided by two times tan of pi over n. For a regular nonagon, n is nine, so apothem equals side divided by two times tangent of pi over nine.
The circumradius follows the sibling identity R equals side divided by two times sine of pi over nine, the perimeter is nine times the side, and the area is one half times perimeter times apothem—or equivalently nine-fourths times side squared over tan of pi over nine. Classic checks include side one, which yields a unit-side regular nonagon apothem near 1.37374, and side two, which scales linearly to about 2.74748.
Because the path uses only multiplication, division, and a single tangent of a fixed angle, it is deterministic, O(1), and free of iterative solvers.
Where nonagon apothem calculations show up in real work
Secondary-school and college geometry courses often ask for the apothem or inradius of a regular nonagon given only the side, and sometimes for circumradius, perimeter, 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 apothem with circumradius or multiplies by nine without the tangent factor.
Architecture, tiling, and manufacturing workflows size nonagonal floor tiles, decorative rosettes, coin-like medallions, and regular nine-sided packing layouts from a measured edge length; apothem feeds center-to-flat offsets while area and perimeter help place cut lists and material estimates. Game, graphics, and CAD libraries occasionally store regular nonagonal cells by side length for maps, strategy boards, and procedural meshes; deterministic inradius and related metrics support collision proxies and layout math without a full computational-geometry stack.
Survey and sketch tools sometimes record flat regular panels as nonagons when only the side is measured in the field; having a shared numeric oracle keeps field notes and office scripts aligned. If you later need irregular nonagon metrics from vertices, area from apothem only, a general regular n-gon with arbitrary side count, or a dedicated octagon or heptagon landing, treat those as separate search intents rather than overloading this nonagon-apothem page.
What you can do with it
Homework and exam checks
Compute regular nonagon apothem from side length, and reject zero or negative sides with a clear error.
Tiles, panels, and decorative nine-gons
From a measured edge length, recover apothem, circumradius, and area for cut lists and material estimates.
FAQ
What formula does this calculator use?
apothem = side / (2 × tan(π/9)) for a regular nonagon. Circumradius uses R = side / (2 × sin(π/9)).
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 apothem the same as circumradius?
No. Apothem (inradius) is the distance from center to a side midpoint (inner circle). Circumradius is the distance from center to a vertex (outer circle). Both are returned so you can compare them.
What else is returned besides apothem?
Circumradius, perimeter (9×side), area (½×perimeter×apothem), sides (9), the apothem 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/nonagon-apothem \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side":4}'const res = await fetch("https://api.kit.forhosting.com/geom/nonagon-apothem", {
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/nonagon-apothem",
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/nonagon-apothem", 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/nonagon-apothem", 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.nonagon_apothem",
"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 nonagon apothem 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 circumradius, perimeter, area, sides count nine, formula_apothem, and method so a single call covers the metrics most worksheets ask for next to the apothem.
What you send and what you get back
Provide one required finite number: side is the length of any edge of the regular nonagon and must be strictly positive. Example checks: side four yields apothem about 5.49495 with perimeter thirty-six; side one yields the unit regular nonagon inradius; 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-tangent 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, apothem, circumradius, perimeter, area, sides set to nine, formula_apothem describing the closed form, and method set to regular_nonagon_side_to_apothem 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 nonagon apothem 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
Nonagon cells in games and CAD
Turn side length into inradius and perimeter for layout, packing, and collision proxies in browser or API.