Hexagon area from apothem
Get the area of a regular hexagon from its apothem, the distance from the centre to a flat side, so apothem 1 gives 2√3 square units. Side length and perimeter follow, and you never place six corners by hand. 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
- 3.4641016151
- Side length
- 1.1547005384
- Perimeter
- 6.9282032303
A regular hexagon is the six-sided polygon with equal sides and equal interior angles—the shape of honeycomb cells, bolts, tiles, 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 six times the square of that apothem by the tangent of pi over six, which equals one over the square root of three, or equivalently two times the square root of three times the square of the apothem. You never have to place six vertices, measure every edge, or split the figure into equilateral triangles by hand.
What regular-hexagon area from the apothem means and how the numbers work
A regular hexagon has six equal sides and six equal interior angles of one hundred twenty 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 six that becomes area equals six times a squared times tan of pi over six. The tangent of thirty degrees has the exact algebraic value one over the square root of three, so the solver uses only a square root and multiplies—no angle-mode ambiguity, no iterative approximation. An equivalent textbook form is two times the square root of three times a squared.
Side length follows as two times a times that same tangent, which is two a over the square root of three; perimeter is six times the side. For a unit apothem the area is six over the square root of three, about three point four six four 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 hexagon search intent and the familiar two-root-three formula many textbooks quote for the apothem path.
What you can do with it
Tiles, honeycomb cells, and signs measured by inradius
Turn a measured center-to-side distance of a regular hexagon into floor, paint, or planting area without reconstructing every edge by hand.
Game, CAD, and mesh area stored by apothem
Validate a regular hexagonal 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 hexagon area tool?
That landing takes side length s. This one takes the apothem (inradius). Same regular hexagon geometry, different measured input and different search intent.
Is this for irregular hexagons?
No. The closed form assumes a regular hexagon (equal sides and equal angles). For arbitrary vertices use a polygon area tool such as the shoelace formula.
Is the hexagon area from apothem calculator free?
The browser tool is free and runs locally.
How do I get a hexagon's area from its apothem?
Multiply two times the square root of three by the apothem squared. It is the polygon rule of half the perimeter times the apothem, simplified because the tangent of π/6 is one over the square root of three.
What else do I get besides the area?
The side length, which is twice the apothem divided by the square root of three, and the perimeter, six times that side. They help you check cuts and edging against your plan.
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/hexagon-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/hexagon-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/hexagon-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/hexagon-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/hexagon-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.hexagon_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 hexagon 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-hexagon 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 hexagon, 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 honeycomb takeoff scripts when only the inradius is known, game-level tooling for six-sided pads stored by apothem, interview-prep sandboxes, or any workflow that needs a reproducible regular hexagon 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 hexagon 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 six, perimeter as six times that side, method set to regular_hexagon_from_apothem, and a short formula block documenting the closed forms including the algebraic tan of pi over six and the equivalent two square-root three 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 over the square root of three, and that area also matches two times the square root of three times apothem squared after the same rounding policy.
Practical tips, limits, and related searches
When teaching, prefer paired examples: apothem one for the pure six-tan constant equal to two times the square root of three; 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 hexagon area capability when available so each search intent stays on its own SEO landing without cannibalizing the other. For flat-to-flat bolt or nut 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 two times the square root of three times a squared by hand for small integers and match the API line by line after rounding. This landing targets people searching for a hexagon area from apothem calculator or regular hexagon inradius area tool; it is not the side-length-only hexagon area calculator, not a general n-gon solver with a variable side count—use the regular polygon calculator when n is not six—and it is not a hexagonal prism volume or hexagonal 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 hexagon with apothem a?
A = 6·a²·tan(π/6) = 2√3·a², with tan(π/6)=1/√3. This calculator evaluates that closed form and also returns side length 2·a·tan(π/6) and perimeter 6s.
What inputs does the hexagon 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.