Pentagram area calculator
A five-pointed star whose tips sit 1 unit from the centre has a filled area of 1.122569941449 square units, against 2.377641290738 for the pentagon on the same circle. Enter the centre-to-tip distance to size star logos, badges or props. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 1
- Area
- 1.1225699414
- Inner radius
- 0.3819660113
- Convex area
- 2.3776412907
- Schlafli
- {5/2}
A regular pentagram is the classic five-pointed star polygon written in Schläfli notation as {5/2}: five equal tips lie on a common circle, and every second vertex is joined so the figure crosses itself into the familiar star. When you know only the circumradius—the distance from the center to any tip—and need the filled area of that star shape, a closed algebraic formula multiplies the square of the radius by a fixed constant built from square roots of five, so you never have to place ten edge segments by hand or run a general star-polygon solver.
What a regular pentagram area formula is and how the numbers work
A regular pentagram is the star polygon {5/2}: its five vertices are equally spaced on a circle of circumradius R, and each edge skips one vertex so the boundary is a self-intersecting isotoxal ten-sided outline with five outer tips and five re-entrant valleys. The filled area of that star—the region you would paint if you filled the silhouette once—depends only on R once regularity is assumed.
The closed form used here is five-eighths times three minus the square root of five, times the square root of ten minus two times the square root of five, all multiplied by R squared. For a unit circumradius the filled area is that algebraic constant itself, about one point one two three square units; scale by R squared for any other positive radius. The response also reports the area of the convex regular pentagon that shares the same circumradius, so students can compare star versus solid five-gon at a glance.
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
Logos, badges, and five-pointed star graphics
Turn a measured tip-to-center distance into filled paint or material area for star emblems without constructing all ten edge segments by hand.
Game, CAD, and mesh area for regular pentagrams
Validate a regular five-pointed star prop or compute silhouette area when only the circumradius is stored in the scene graph or design file.
FAQ
What is the area of a regular pentagram with circumradius R?
A = (5/8)×(3−√5)×√(10−2√5)×R² for the filled star. This calculator also returns the inner valley radius R(3−√5)/2 and the convex pentagon area for the same R.
Is this the same as a regular pentagon area calculator?
No. A pentagon is the convex five-gon; a pentagram is the star polygon {5/2}. Same circumradius yields a larger convex area than the filled star area.
How does this differ from a general star polygon area tool?
This landing is specialized to the regular five-pointed star from a single circumradius for pentagram-area searches. A general {n/d} tool accepts variable points and density for other stars.
Is the pentagram area calculator free?
The browser tool is free and runs locally.
What is the inner radius of a pentagram?
It is the distance from the centre to the five inner corners between the points. For a tip distance of 1 it is 0.38196601125, and it always scales in step with the outer radius.
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/pentagram-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":1}'const res = await fetch("https://api.kit.forhosting.com/geom/pentagram-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/pentagram-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/pentagram-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/pentagram-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
{
"radius": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.pentagram_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 pentagram area calculator accepts a single positive circumradius, returns the filled star area together with the inner valley radius and the convex regular-pentagon area for the same radius, and rejects zero or negative radii 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, design takeoffs, game-level tooling, and production pipelines all see the same deterministic JSON every time you send the same radius.
What a regular pentagram area formula is and how the numbers work
That constant is exact in the algebraic sense; floating-point evaluation uses a few square roots and multiplies, which is enough for homework, drafting, and API clients that need stable twelve-decimal rounding. Degenerate inputs—zero, negative, missing, or non-finite radius—do not form a positive finite regular pentagram, 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, logo and badge design takeoffs, game-level tooling for five-pointed stars, interview-prep sandboxes, or any workflow that needs a reproducible regular pentagram area without installing a computer-algebra package on every machine. Send a JSON body with one required field. Field radius is the positive finite circumradius of the regular pentagram—the tip-to-center distance; aliases R, r, and circumradius are accepted so form UIs and scripts can keep their preferred names. On success the response echoes radius, returns area as the filled star region, inner_radius as the distance from center to a valley of the isotoxal outline, convex_area as the regular pentagon area for the same R, schlafli set to the string {5/2}, method set to regular_pentagram_closed_form, and a short formula block documenting the closed forms. Failed validation—missing the field, a non-finite value, a non-positive radius, or a radius 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 radius one and radius two with a deliberate zero or negative radius to verify clients handle invalid_input cleanly and never treat a failed solve as a charged success.
Practical tips, limits, and related searches
When teaching, prefer paired examples: radius one for the pure algebraic constant; radius two so students can verify that area scales with R squared (four times the unit area); and radius zero or negative to demonstrate the hard rejection path that protects callers from non-positive geometry. Keep the circumradius 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 circumradius, 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 a pentagram area calculator or regular five-pointed star area tool from circumradius; it is not a convex pentagon area tool (use the pentagon area calculator when you have side length of a solid five-gon), and it is not a general star-polygon solver with variable point count and density—use the star polygon area calculator when n and d are not five and two. Prefer this endpoint when the figure is known to be a regular pentagram and only the tip radius is measured. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible radii while learning the contract. Round-trip checks are easy: after solving, confirm that area scales with the square of radius, that inner_radius equals radius times three minus root five over two, and that convex_area is larger than the star area for the same R within floating tolerance.
Integration use cases
Homework and exam checks for regular pentagram area
Give students a circumradius; compare their hand-computed closed-form star area and inner radius against a deterministic API or free browser result.
Technical questions
What inputs does the pentagram area calculator need?
Only one positive finite number: radius (aliases R, r, circumradius)—the tip-to-center distance. Zero, negative, missing, or non-finite values return invalid_input.