ForHosting KIT · Math

Hexagon apothem calculator

Turn a hexagon's edge length into its inner radius, so a side of 2 gives an apothem of √3, about 1.732050807569. It is the clearance you need for tiles, nuts and honeycomb cells. Free in your browser.

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Worked example

If you enter

Side: 2

You get

Apothem: 1.7320508076

The apothem of a regular hexagon is the perpendicular distance from the centre to the middle of any side—the radius of the inscribed circle, also called the inradius. Designers, fabricators, teachers, and game developers often know the edge length of a hexagonal tile, nut, honeycomb cell, or floor plate and need that inner radius for clearances, packing density, or area formulas. This hexagon apothem calculator accepts a single positive side length, applies the closed form a equals the square root of three over two times the side, and returns the apothem together with a short formula string you can show next to the answer.

What the apothem of a regular hexagon is and why the formula is simple

A regular hexagon has six equal sides and six equal interior angles of one hundred twenty degrees. Connecting the centre to every vertex partitions the hexagon into six equilateral triangles, each with side equal to the hexagon side length s. The apothem is the height of one of those equilateral triangles from the centre down to the base (a side of the hexagon).

The height of an equilateral triangle with side s is exactly the square root of three over two times s, so the apothem is a equals (√3/2) times s. That length is also the inradius: the radius of the unique circle that touches all six sides from the inside. It is not the same as the circumradius (vertex radius), which for a regular hexagon equals s itself.

Students sometimes confuse the two; this tool reports only the apothem so the search intent for hexagon apothem calculator stays crisp. Equivalent trigonometry confirms the same identity: a equals s times cosine of thirty degrees, or a equals s divided by two times the tangent of thirty degrees (the general regular n-gon apothem formula with n equals six). All three expressions are algebraically identical for a perfect hexagon, and the implementation uses the square-root form for a single deterministic floating-point path.

What you enter and what you get back

Zero and negative side lengths also fail, because a geometric edge length must be strictly greater than zero—there is no meaningful apothem for a collapsed or inverted hexagon. On success the body echoes side, returns apothem rounded to twelve decimal places for stable goldens across runtimes, and includes a formula string restating a equals (√3/2)·s as the inradius of a regular hexagon. Example: side equals two yields apothem equal to the square root of three, about one point seven three two zero five zero eight zero seven five seven.

Side equals one yields apothem equal to half the square root of three. Units are not converted: if you pass centimetres you get centimetres, if you pass inches you get inches.

Where a hexagon apothem calculator earns its keep

Architecture and interior design use regular hexagonal tiles and columns; knowing the inradius from the published edge length sets centre-to-side clearances and inscribed circular fittings. Manufacturing of hexagonal stock, bolt heads, and honeycomb panels needs the same distance when specifying inscribed-circle gauges or when nesting hex blanks inside circular stock. Game and map engines that place hexagonal grids often store side length as the canonical size and must convert to apothem when aligning flat-to-flat spacing (flat-to-flat equals twice the apothem).

Keep requests small, pin the rounded apothem in tests, and you get reproducible inradii for homework, LMS plugins, CAD macros, and edge workers without standing up a general polygon library when the only figure that matters is the regular hexagon.

What you can do with it

Hexagonal tile and panel clearances

Convert a published hex tile edge length into the inradius so fittings, holes, and centre marks sit correctly relative to each flat side.

Classroom and exam checks

Let students enter side length s and verify a=(√3/2)·s against textbook equilateral-triangle height methods without calculator slip-ups.

Hex grid and game layout tools

Derive flat-to-flat spacing (twice the apothem) from a side-based hex grid size for map engines and procedural level generators.

FAQ

What happens if the side length is zero or negative?

Side length must be strictly greater than zero; non-positive values are not billed.

Is the apothem the same as the circumradius?

No. For a regular hexagon the circumradius (centre to vertex) equals the side length s, while the apothem (centre to side midpoint) is (√3/2)·s.

Which units does the calculator use?

None are assumed. Output uses the same unit as the input side length—millimetres in, millimetres out.

What is the formula for the apothem of a hexagon?

Multiply the side by the square root of three and divide by two. The same value comes from the side times the cosine of 30 degrees, because the apothem is the height of one of six equilateral triangles.

Can I get the hexagon's area from the apothem?

Yes, by hand: the area is three times the side times the apothem. This tool returns only the apothem, so the area and perimeter are left to you or to a dedicated area calculator.

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

POSThttps://api.kit.forhosting.com/geom/hexagon-apothem

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-apothem \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"side":2}'

const res = await fetch("https://api.kit.forhosting.com/geom/hexagon-apothem", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "side": 2
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/geom/hexagon-apothem",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "side": 2
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/hexagon-apothem", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"side":2}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"side":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/hexagon-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": 2
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "geom.hexagon_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

Per request$0.002

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

Non-positive or non-finite side values fail with invalid_input instead of a silent NaN.

Input contract, response shape, and validation rules

POST a JSON body with one required field: side, a finite positive number giving the edge length of the regular hexagon. The value may arrive as a JSON number or as a numeric string; empty strings, missing fields, non-numeric text, NaN, and Infinity fail with invalid_input naming the field. Pair the response with your own area formula if needed—area of a regular hexagon is (3√3/2) s squared, or equivalently three times s times a—this endpoint derives the apothem only and does not emit area, perimeter, or drawings. Failed validation does not consume prepaid balance; successful parse-tier calls use the published base price.

Technical questions

How do I find the apothem of a regular hexagon from the side length?

Send a finite positive number side. The API returns apothem a = (√3/2)·s, the inradius of the regular hexagon, plus a short formula string.

What does the hexagon apothem calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

Run — free