Octagon area from apothem
Measure 1 unit from the center of a regular octagon to any side and its area is 8(√2 − 1), about 3.313708498985 square units. Give that apothem and get the area plus the side and perimeter it implies, with no edge to measure. 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
- 3.313708499
- Side length
- 0.8284271247
- Perimeter
- 6.627416998
A regular octagon is the eight-sided polygon with equal sides and equal interior angles—the shape of stop signs, gazebo footprints, classroom figures, and many CAD pads. When you know the apothem (the perpendicular distance from the center to any side) and need the enclosed area, the closed form multiplies eight times the square of that apothem by the tangent of pi over eight, which equals the square root of two minus one. You never have to place eight vertices, measure every edge, or split the figure into isosceles triangles by hand.
What regular-octagon area from the apothem means and how the numbers work
A regular octagon has eight equal sides and eight equal interior angles of one hundred thirty-five 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 eight that becomes area equals eight times a squared times tan of pi over eight. The tangent of twenty-two and a half degrees has the exact algebraic value square root of two minus one, so the solver uses only a square root and multiplies—no angle-mode ambiguity, no iterative approximation. An equivalent textbook form is eight times the square root of two minus one times a squared.
Side length follows as two times a times that same tangent, which is two a times the square root of two minus one; perimeter is eight times the side. For a unit apothem the area is eight times the square root of two minus one, about three point three one three 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 need for vertex coordinates in the input contract.
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 octagon search intent and the familiar eight-times-root-two-minus-one formula many textbooks quote for the apothem path.
What you can do with it
Stop signs, gazebos, and pads measured by inradius
Turn a measured center-to-side distance of a regular octagon into floor, paint, or planting area without reconstructing every edge by hand.
Game, CAD, and mesh area stored by apothem
Validate a regular octagonal pad or compute surface area when the scene graph stores the inradius rather than side length.
FAQ
How is this different from a side-length octagon area tool?
That landing takes side length s. This one takes the apothem (inradius). Same regular octagon geometry, different measured input and different search intent.
Is this for irregular octagons?
No. The closed form assumes a regular octagon (equal sides and equal angles). For arbitrary vertices use a polygon area tool such as the shoelace formula.
Is the octagon area from apothem calculator free?
The browser tool is free and runs locally.
What is the area formula when the apothem is known?
Area equals eight times the apothem squared times the tangent of π/8. Because that tangent is exactly √2 − 1, the same rule reads eight times √2 − 1 times the apothem squared, which is how many textbooks quote it.
What if the apothem is zero or negative?
You get a clear refusal instead of a result. No regular octagon has a center-to-side distance of zero or less, so returning an area of zero would only hide a measuring or typing slip.
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-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/octagon-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/octagon-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/octagon-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/octagon-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.octagon_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 octagon area from apothem calculator accepts a single positive apothem, returns 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, and production pipelines all see the same deterministic JSON every time.
What regular-octagon 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 octagon, 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, tiling or gazebo takeoff scripts when only the inradius is known, game-level tooling for eight-sided pads stored by apothem, interview-prep sandboxes, or any workflow that needs a reproducible regular octagon 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 octagon 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 eight, perimeter as eight times that side, method set to regular_octagon_from_apothem, and a short formula block documenting the closed forms including the algebraic tan of pi over eight and the equivalent eight times square-root-of-two-minus-one times a squared expression. 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 the square root of two minus one, and that area also matches eight 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 eight-tan constant equal to eight times the square root of two minus one; 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 the companion side-based octagon area capability so each search intent stays on its own SEO landing without cannibalizing the other. For flat-to-flat bolt, nut, or sign specs, 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 eight times the square root of two minus one times a squared by hand for small integers and match the API line by line after rounding. This landing targets people searching for an octagon area from apothem calculator or regular octagon inradius area tool; it is not the side-length-only octagon area calculator, not a general n-gon solver with a variable side count—use the regular polygon calculator when n is not eight—and it is not an octagonal prism volume or octagonal 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 area, side length, and perimeter against a deterministic API or free browser result.
Technical questions
What is the area of a regular octagon with apothem a?
A = 8·a²·tan(π/8) = 8(√2−1)·a², with tan(π/8)=√2−1. This calculator evaluates that closed form and also returns side length 2·a·tan(π/8) and perimeter 8s.
What inputs does the octagon 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.