Pentagon area from apothem
A regular pentagon whose centre sits 1 unit from each side covers 3.632712640027 square units. Enter that centre-to-side distance, the apothem, to get the area plus side length and perimeter, without measuring a single edge. 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
- 3.63271264
- Side length
- 1.453085056
- Perimeter
- 7.2654252801
A regular pentagon is the five-sided polygon with equal sides and equal interior angles—the shape of many badges, garden beds, classroom figures, and 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 five times the square of that apothem by the tangent of pi over five, evaluated algebraically as the square root of five minus two times the square root of five. You never have to place five vertices, measure every edge, or split the figure into isosceles triangles by hand.
What regular-pentagon area from the apothem means and how the numbers work
A regular pentagon has five equal sides and five equal interior angles of one hundred eight 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 five that becomes area equals five times a squared times tan of pi over five. The tangent of thirty-six degrees has the exact algebraic value square root of five minus two times the square root of five, so the solver uses only square roots and multiplies—no angle-mode ambiguity, no iterative approximation. Side length follows as two times a times that same tangent; perimeter is five times the side.
For a unit apothem the area is five times tan of pi over five, about three point six 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 height-to-side is the measured quantity rather than an edge length.
What you can do with it
Garden beds and signs measured by inradius
Turn a measured center-to-side distance of a regular pentagon into floor, paint, or planting area without reconstructing every edge by hand.
Game, CAD, and mesh area stored by apothem
Validate a regular pentagonal pad or compute surface area when the scene graph stores the inradius rather than side length.
FAQ
How is this different from the side-length pentagon area tool?
That landing takes side length s. This one takes the apothem (inradius). Same regular pentagon geometry, different measured input and different search intent.
Is this for irregular pentagons?
No. The closed form assumes a regular pentagon (equal sides and equal angles). For arbitrary vertices use a polygon area tool such as the shoelace formula.
Is the pentagon area from apothem calculator free?
The browser tool is free and runs locally.
What exactly is the apothem of a pentagon?
It is the perpendicular distance from the centre of the regular pentagon to the midpoint of any side. It also equals the radius of the circle drawn inside the pentagon that touches all five sides.
Can I get the side length from the apothem too?
Yes. The side length and the perimeter come with the area: for an apothem of 1 the side is 1.453085056011 and the perimeter 7.265425280054, so one measurement rebuilds the whole pentagon.
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/pentagon-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/pentagon-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/pentagon-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/pentagon-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/pentagon-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.pentagon_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 pentagon 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-pentagon 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 pentagon, 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 garden-bed takeoff scripts when only the inradius is known, game-level tooling for five-sided pads stored by apothem, interview-prep sandboxes, or any workflow that needs a reproducible regular pentagon 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 pentagon 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 five, perimeter as five times that side, method set to regular_pentagon_from_apothem, and a short formula block documenting the closed forms including the algebraic tan of pi over five. 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, and that side_length equals two times apothem times the documented tangent constant.
Practical tips, limits, and related searches
When teaching, prefer paired examples: apothem one for the pure five-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 the companion side-based pentagon area capability so each search intent stays on its own SEO landing without cannibalizing the other. 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 five times a squared times square root of five minus two square roots of five by hand for small integers and match the API line by line after rounding. This landing targets people searching for a pentagon area from apothem calculator or regular pentagon inradius area tool; it is not the side-length-only pentagon area calculator, not a general n-gon solver with a variable side count—use the regular polygon calculator when n is not five—and it is not a pentagonal prism volume 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 pentagon with apothem a?
A = 5·a²·tan(π/5), with tan(π/5)=√(5−2√5). This calculator evaluates that closed form and also returns side length 2·a·tan(π/5) and perimeter 5s.
What inputs does the pentagon 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.