Regular polygon central angle calculator
Enter the number of sides of a regular polygon and get the angle at its center between two neighbouring corners: a regular pentagon has a central angle of 72°, a fifth of a full turn of 360°. Free in your browser.
- Free
- Runs in your browser
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- Your data stays on your device
Run — free
Worked example
- N
- 5
- Central angle deg
- 72
A regular polygon is a closed planar shape with n equal sides and n equal vertex angles, all vertices lying on a common circle. The central angle is the angle formed at that circumcenter by two radii drawn to adjacent vertices: it is the slice of a full turn that belongs to one side. Students, teachers, CAD authors, and game-tooling authors need this measure when they place vertices on a circle, label sector diagrams, or check homework for triangles, squares, pentagons, and higher n-gons. Send an integer n of three or more and receive the central angle in degrees plus a short formula string. No trigonometry tables, no floating-point hand errors for common classroom cases, and the same pure logic runs free in the browser before you wire the paid automation path for batch checks and scripts.
What the central angle of a regular polygon means
Imagine the circumcircle of a regular n-gon and the center of that circle. Each vertex of the polygon sits on the circle, spaced evenly around a full rotation of three hundred sixty degrees. Connecting the center to two consecutive vertices draws an isosceles triangle whose apex at the center is the central angle for one side.
Because the vertices are equally spaced, every such apex is identical, and the n central angles sum to exactly one full turn. Therefore each central angle is three hundred sixty degrees divided by n. An equilateral triangle has central angle one hundred twenty degrees; a square has ninety; a regular pentagon has seventy two; a regular hexagon has sixty; a regular octagon has forty five.
In many textbooks this central angle is also the angle subtended by one side at the center, and it equals the exterior angle of a regular n-gon when exterior angles are defined as the turn of a boundary walk.
Where the central angle is put to use
Classrooms use a regular polygon central angle calculator so students can verify sector diagrams and circle-based constructions for heptagons and dodecagons without memorizing a long special-case table. CAD and architecture scripts need the central step when they place columns, bolts, or decorative tiles on a circular layout with equal angular spacing.
Game and graphics pipelines walk a turtle or rotate a matrix by three hundred sixty over n when building regular stars, radar charts, UI badges, and procedural props; pinning that step in a deterministic service keeps tests green across platforms. Documentation sites and SEO landings that teach regular-polygon formulas benefit from a dedicated page for central angle rather than burying the identity inside a multi-output angle tool. Failed validation does not consume balance.
What you can do with it
Classroom central-angle checks
Let students request the central angle for n from three upward and verify that each sector is exactly 360°/n against textbook tables and diagram labels.
Circular layout and CAD spacing
Space bolts, columns, or decorative vertices evenly around a circle by stepping the known central angle for the chosen regular n-gon.
Procedural regular shapes in games and UI
Drive rotation steps of 360/n when generating regular polygons for charts, badges, or game props with deterministic, testable angles.
FAQ
How do I calculate the central angle of a regular polygon?
The central angle in degrees is 360/n. That is the angle at the circumcenter between two adjacent vertices.
What happens if n is below 3?
A polygon needs at least three sides, so n less than three is rejected without charge.
Is the central angle returned in degrees or radians?
Convert to radians client-side by multiplying by π/180 if your pipeline expects radians.
Is the central angle the same as the exterior angle?
For a regular n-gon, the central angle 360°/n equals each exterior turn angle of a boundary walk.
How do I use the central angle to space points around a circle?
Step around the center by the central angle each time. For five equally spaced points, such as columns or bolts on a circular layout, each one sits 72° from the next.
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/regular-polygon-central-angle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"n":5}'const res = await fetch("https://api.kit.forhosting.com/geom/regular-polygon-central-angle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"n": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/regular-polygon-central-angle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"n": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/regular-polygon-central-angle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"n":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"n":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/regular-polygon-central-angle", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"n": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.regular_polygon_central_angle",
"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 regular polygon central angle calculator turns the single identity central angle equals three hundred sixty degrees divided by n into a deterministic API and a free browser tool.
What the central angle of a regular polygon means
This endpoint reports the central angle in degrees only, which matches school materials, drafting UIs, and most homework answer keys. If you later need radians, multiply degrees by pi over one hundred eighty on the client; keeping the API in degrees avoids converting twice in education-first workflows.
Input contract, formula, and error behaviour
Common classroom values such as n equals three, four, five, six, eight, and twelve produce clean terminating decimals that match textbook tables. Pair it with your own drawing code or with other KIT geometry tools when you need a full regular-polygon description. POST a JSON body with a single required field n: an integer greater than or equal to three and at most one million. Values below three fail with invalid_input because a polygon needs at least three sides; non-integers, missing fields, empty strings, and n above the published maximum also fail with invalid_input and are not billed. On success the body contains n, central_angle_deg, and a short formula string of the form central_angle equals three hundred sixty over n with the concrete n echoed. Bound n, keep the request small, and treat the response as pure geometry data: this capability does not emit vertex coordinates, radii, or interior angles.
Where a central-angle API earns its keep
Use the free browser path for one-off checks and the API for homework generators, design validation, and mesh tooling that need the same number every time.
Technical questions
What does the regular polygon central angle calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.