Octagon area calculator
A regular octagon with sides of 1 encloses 2(1 + √2), about 4.828427124746 square units, and the area grows with the square of the side. Enter one side length and get the area, perimeter and apothem. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side length
- 1
- Area
- 4.8284271247
- Perimeter
- 8
- Apothem
- 1.2071067812
A regular octagon is the eight-sided polygon whose sides are equal and whose interior angles are all equal—the familiar stop-sign shape, the footprint of many gazebos and game boards, and a staple of classroom geometry worksheets. When you know only the side length and need the enclosed area, the classic closed form multiplies the square of that length by two times one plus the square root of two, so you never have to place vertices, measure an apothem by hand, or split the figure into triangles yourself.
What a regular octagon area formula is and how the numbers work
A regular octagon has eight equal sides and eight equal interior angles of one hundred thirty-five degrees. Its area depends only on the side length s: there is no second independent dimension once regularity is assumed. The standard closed form is two times one plus the square root of two, all multiplied by s squared.
An equivalent trigonometric form is two times s squared over the tangent of pi over eight—the regular n-gon formula specialized to n equals eight. Because tan of pi over eight equals square root of two minus one, rationalizing recovers the same two-times-one-plus-root-two factor. For a unit side, the area is that algebraic constant itself, about four point eight two eight square units; scale by s squared for any other positive side.
Perimeter is simply eight times s. Because area equals one-half times perimeter times apothem, the apothem follows as two times area over perimeter without a separate angle library. 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.
What you can do with it
Gazebos, stop-sign panels, and eight-sided floor plans
Turn a measured regular-octagon edge into floor, paint, or material area without constructing an apothem or placing eight vertices by hand.
Game, CAD, and mesh area for regular eight-gons
Validate a regular octagonal pad or compute surface area when only the side length is stored in the scene graph or design file.
FAQ
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.
How does this differ from a regular polygon calculator?
This landing is specialized to eight equal sides from a single side length for octagon-area searches. A general regular n-gon tool accepts a variable side count for other polygons.
Is the octagon area calculator free?
The browser tool is free and runs locally.
Why is the octagon area 2(1 + √2) times the side squared?
The rule for a regular eight-sided polygon is twice the side squared divided by the tangent of π/8. That tangent is exactly √2 − 1, and simplifying the fraction turns it into the factor 2(1 + √2).
What happens if the side length is zero or negative?
It is rejected with a clear message. A zero or negative side does not form a real octagon, 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/octagon-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side_length":1}'const res = await fetch("https://api.kit.forhosting.com/geom/octagon-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side_length": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/octagon-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side_length": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/octagon-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"side_length":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side_length":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/octagon-area", 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_length": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.octagon_area",
"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 calculator accepts a single positive side length, returns the area together with perimeter and apothem for cross-checks, and rejects zero or negative lengths 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, CAD takeoffs, and production pipelines all see the same deterministic JSON every time.
What a regular octagon area formula is and how the numbers work
That constant is exact in the algebraic sense; floating-point evaluation uses a single square root and a few multiplies, which is enough for homework, drafting, and API clients that need stable twelve-decimal rounding. Degenerate inputs—zero, negative, missing, or non-finite side length—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, landscaping or gazebo takeoff scripts, game-level tooling for eight-sided pads, 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 side_length is the positive finite side length of the regular octagon; aliases side, s, a, and length are accepted so form UIs and scripts can keep their preferred names. On success the response echoes side_length, returns area as the enclosed region, perimeter as eight times the side, apothem as the inradius-style distance from center to a side, method set to regular_octagon_closed_form, and a short formula block documenting the closed forms. Failed validation—missing the field, a non-finite value, a non-positive side, or a side 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 side one and side ten with a deliberate zero or negative side to verify clients handle invalid_input cleanly.
Practical tips, limits, and related searches
When teaching, prefer paired examples: side length one for the pure algebraic constant; side length two so students can verify that area scales with s squared (four times the unit area); and side length zero or negative to demonstrate the hard rejection path that protects callers from non-positive geometry. Keep the side length 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 side, students can recompute the constant by hand for small integers and match the API line by line after rounding. This landing targets people searching for an octagon area calculator or regular octagon side length tool; it is 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 octahedron volume tool for three-dimensional solids. Prefer this endpoint when the figure is known regular and only the side is measured. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible sides while learning the contract. Round-trip checks are easy: after solving, confirm perimeter equals eight times side_length, and that area equals one-half times perimeter times apothem within floating tolerance.
Integration use cases
Homework and exam checks for regular octagon area
Give students a side length; compare their hand-computed closed-form area, perimeter, and apothem against a deterministic API or free browser result.
Technical questions
What is the area of a regular octagon with side length s?
A = 2×(1+√2)×s². This calculator evaluates that closed form and also returns perimeter 8s and apothem 2A/P.
What inputs does the octagon area calculator need?
Only one positive finite number: side_length (aliases side, s, a, length). Zero, negative, missing, or non-finite values return invalid_input.