Nonagon area from apothem
A regular nonagon whose center sits 1 unit from each side has an area of 9·tan(π/9), about 3.275732108396 square units. Enter the measured apothem and get the area, side and perimeter without placing nine vertices. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Apothem
- 1
- Area
- 3.2757321084
- Side length
- 0.7279404685
- Perimeter
- 6.5514642168
A regular nonagon—also called an enneagon—is the nine-sided polygon with equal sides and equal interior angles. When you already know the apothem (the perpendicular distance from the center to any side, equal to the inradius of the inscribed circle) and need the enclosed area, the standard regular-polygon identity multiplies nine times the square of that apothem by the tangent of pi over nine. You do not need nine vertex coordinates, a full side-length survey, or a general computer-algebra session.
What regular-nonagon area from the apothem means and how the numbers work
A regular nonagon has nine equal sides and nine equal interior angles of one hundred forty degrees. Once the figure is known to be regular, a single positive apothem a fixes the entire shape up to similarity scaling. The standard area identity for any regular n-gon is one-half times perimeter times apothem.
For n equals nine that becomes area equals nine times a squared times tan of pi over nine. The central half-angle is twenty degrees, so the solver evaluates tan of pi over nine once and multiplies—no angle-mode ambiguity, no iterative approximation, and no need to place nine vertices in the plane. Side length follows as two times a times that same tangent; perimeter is nine times the side.
For a unit apothem the area is nine times tan of pi over nine, roughly three point two seven six square units; scale by a squared for any other positive apothem. Every step is elementary arithmetic on one scalar: no iteration, no sampling, no wall-clock dependence, and no dependence on floating Date or network state. Compared with a side-length-only landing, this path matches surveys and drafting notes where the inradius or flat-to-flat half-distance is the measured quantity rather than an edge length.
Compared with a general n-gon calculator, this landing is dedicated to the nonagon and enneagon search intent and the familiar nine-times-a-squared-times-tan form textbooks quote for the apothem path when n is nine.
What you can do with it
Decorative layouts and planters measured by inradius
Turn a measured center-to-side distance of a regular nonagon into floor, paint, planting, or fabric area without reconstructing every edge by hand.
Game, CAD, and mesh area stored by apothem
Validate a regular enneagon pad or compute surface area when the scene graph stores the inradius rather than side length.
FAQ
Is a nonagon the same as an enneagon?
Yes. Nonagon (from Latin) and enneagon (from Greek) both mean a nine-sided polygon. This tool assumes a regular nonagon—equal sides and equal angles.
How is this different from a side-length nonagon area tool?
A side-length tool takes edge length s. This one takes the apothem (inradius). Same regular nonagon geometry, different measured input and different search intent.
Is the nonagon area from apothem calculator free?
The browser tool is free and runs locally.
What is the area formula from the apothem?
Area equals nine times the apothem squared times the tangent of π/9. It comes from the rule for any regular polygon, half the perimeter times the apothem, with the side written as twice the apothem times that same tangent.
What happens if the apothem is zero or negative?
It is rejected with a clear message. A zero or negative distance from the center to a side does not describe a real nonagon, so you get an error instead of an area of zero or a meaningless number.
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-area-apothem \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"apothem":1}'const res = await fetch("https://api.kit.forhosting.com/geom/nonagon-area-apothem", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"apothem": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/nonagon-area-apothem",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"apothem": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/nonagon-area-apothem", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"apothem":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"apothem":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/nonagon-area-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
{
"apothem": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.nonagon_area_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
This nonagon area from apothem calculator accepts a single positive apothem, returns the area together with side length and perimeter for cross-checks, and rejects zero or negative apothems with invalid_input instead of inventing a meaningless number. The same pure solve function powers the free browser tool and the prepaid API path, so homework checks, grading scripts, landscaping takeoffs, decorative-layout estimators, and production pipelines all see the same deterministic JSON every time.
What regular-nonagon area from the apothem means and how the numbers work
Its apothem is the inradius: the radius of the inscribed circle, equal to the perpendicular distance from the geometric center to the midpoint of any side. Degenerate inputs—zero, negative, missing, or non-finite apothem—do not form a positive finite regular nonagon, so this tool returns invalid_input rather than zero or NaN.
How to call the API and what comes back
Use it from geometry homework pipelines, decorative nine-sided layout or planter takeoff scripts when only the inradius is known, game-level tooling for enneagon pads stored by apothem, interview-prep sandboxes, or any workflow that needs a reproducible regular nonagon area without installing a computer-algebra package on every machine. Send a JSON body with one required field. Field apothem is the positive finite distance from the center of the regular nonagon to any side; aliases a, r, and inradius are accepted so form UIs and scripts can keep their preferred names. On success the response echoes apothem, returns area as the enclosed region, side_length derived from two times apothem times tan of pi over nine, perimeter as nine times that side, method set to regular_nonagon_from_apothem, and a short formula block documenting the closed forms including the general n-gon pattern and the specialized nine-side expressions. Failed validation—missing the field, a non-finite value, a non-positive apothem, or an apothem outside published limits—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Pair success cases such as apothem one and apothem two with a deliberate zero or negative apothem to verify clients handle invalid_input cleanly. Round-trip checks are easy: after solving, confirm area equals one-half times perimeter times apothem within floating tolerance, that side_length equals two times apothem times tan of pi over nine, and that area also matches nine times that same factor times apothem squared after the same rounding policy.
Practical tips, limits, and related searches
When teaching, prefer paired examples: apothem one for the pure nine-tan constant; apothem two so students can verify that area scales with a squared (four times the unit-apothem area); and apothem zero or negative to demonstrate the hard rejection path that protects callers from non-positive geometry. If you already know the side length instead, use a companion side-based regular-polygon area path so each search intent stays on its own SEO landing without cannibalizing the other. For flat-to-flat decorative tiles, hardware blanks, or playground figures, remember that the full width across flats is twice the apothem, so half of that measured distance is the number you pass here. Keep the apothem inside the published positive range up to one trillion so intermediate products stay finite in IEEE double precision on the Worker and in the browser. Units are whatever you pass—metres, feet, or game units—as long as you interpret area in the square of that unit; the API does not convert units for you. Because the formula only needs the apothem, students can recompute nine times tan of pi over nine times a squared by hand for small integers and match the API line by line after rounding. This landing targets people searching for a nonagon area from apothem calculator, enneagon inradius area tool, or regular nine-sided polygon area from apothem; it is not the side-length-only nonagon area calculator, not a general n-gon solver with a variable side count—use the regular polygon calculator when n is not nine—and it is not a nonagonal prism volume or nonagonal pyramid tool. Prefer this endpoint when the figure is known regular and only the apothem or inradius is measured. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible apothems while learning the contract.
Integration use cases
Homework and exam checks from a measured apothem
Give students a positive apothem; compare their hand-computed nonagon area, side length, and perimeter against a deterministic API or free browser result.
Technical questions
What is the area of a regular nonagon with apothem a?
A = 9·a²·tan(π/9). This calculator evaluates that closed form and also returns side length 2·a·tan(π/9) and perimeter 9s.
What inputs does the nonagon area from apothem calculator need?
Only one positive finite number: apothem (aliases a, r, inradius). Zero, negative, missing, or non-finite values return invalid_input.