Circular sector centroid calculator
Find where a pie-slice shape balances: for a quarter circle of radius 5, the centroid sits just over 3 units from the center, out along the bisector of the angle. Degrees or radians both work. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 5
- Angle
- 90
- Angle unit
- Degrees
- Angle radians
- 1.5707963268
- Angle degrees
- 90
- Centroid distance
- 3.0010543872
A circular sector is the pie-slice region bounded by two radii and the included arc. Its centroid — the geometric center of mass of a uniform lamina — does not sit at the circle center; it lies farther out along the angle bisector. Given a positive radius and a non-zero central angle in degrees or radians, this calculator returns that centroid distance from the center using the closed-form formula d equals four r sine of half-theta over three theta.
How the sector centroid distance is derived
Place the sector symmetrically about the positive x-axis so the bisector is the x-axis and the polar angle runs from minus theta over two to plus theta over two. Area of the sector is one half r squared times absolute theta in radians.
The first moment integral of x equals rho cosine phi over the polar domain reduces to a product of a radial integral of rho squared and an angular integral of cosine, which evaluates to four r sine of half absolute theta over three absolute theta after dividing by area. That closed form is the distance d returned here.
Classic checks anchor the implementation: a semicircle of radius r at theta equals pi yields the familiar four r over three pi; a full disk at two pi yields sine of pi equals zero so d equals zero at the center; a quarter-circle at ninety degrees yields four r times sine of pi over four over three times pi over two.
Absolute value of theta keeps d non-negative when a signed orientation angle is flipped, while the response still echoes the original signed angle in both degrees and radians for audit trails. Because the path uses only multiplication, division, and a single sine after the pi conversion, it is deterministic, O of one, and free of iterative solvers.
Where circular sector centroids show up in real work
Statics and strength-of-materials courses constantly ask for the centroid of a circular sector or semicircular area when computing first moments, composite centers of gravity, and hydrostatic force locations; a trusted calculator with a strict zero-angle gate speeds homework grading and removes spreadsheet mistakes where someone uses two r over three from a triangle instead of the sector formula.
Manufacturing and drafting workflows use sector-shaped cutouts, cam lobes, and pie-slice laminas; knowing how far the mass center sits from the pivot keeps balance calculations and fastener layouts aligned with the same four-r-sine formula. Game, graphics, and UI libraries often store pie charts, skill cooldowns, and cone regions as polar wedges; the centroid distance feeds label placement, balance points, and simple physics proxies without a full computational-geometry stack.
If you later need sector area and perimeter, segment area from chord height, or a triangle centroid from three vertices, treat those as separate search intents rather than overloading this circular-sector-centroid landing.
What you can do with it
Statics homework and exam checks
Compute sector centroid distance from radius and degrees or radians, and reject a zero central angle with a clear error.
Balance of sector-shaped parts
From a positive radius and central angle, recover how far the mass center sits from the circle center along the bisector.
FAQ
What formula does this calculator use?
Degrees are converted with θ_rad = angle × π / 180.
What happens if the central angle is zero?
A zero angle is a degenerate sector and the formula divides by the angle; the error message names the field and the reason.
What does it cost?
The browser tool is free and runs locally.
Where is the centroid of a semicircle?
It lies on the line of symmetry, at a distance of four times the radius divided by three pi from the center. That is the classic value used for semicircular areas in statics.
Why is the centroid of a full circle at its center?
At a full turn the sine of half the angle is zero, so the distance becomes zero. The balance point of a whole disk is its center, just as symmetry suggests.
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/sector-centroid \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":5,"angle":90}'const res = await fetch("https://api.kit.forhosting.com/geom/sector-centroid", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 5,
"angle": 90
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/sector-centroid",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 5,
"angle": 90
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/sector-centroid", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":5,"angle":90}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":5,"angle":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/sector-centroid", 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": 5,
"angle": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.sector_centroid",
"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
A zero central angle is rejected with a clear invalid_input error because the formula divides by the angle and a zero angle is a degenerate sector, not a silent success. No network calls, no random sampling, and no floating-point surprises beyond ordinary IEEE-754 arithmetic on the numbers you provide.
How the sector centroid distance is derived
Huge radius or angle magnitudes can still overflow IEEE-754 double range; when the result is non-finite the request fails with a clear overflow message instead of returning Infinity in JSON.
What you send and what you get back
Provide two required finite numbers: radius must be strictly positive, and angle is the central angle of the sector and must be non-zero. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation without reimplementing the polar moment formula in every language. Optionally set angle_unit to degrees, the default, or radians so the same numeric angle can be interpreted either way. Missing fields, empty strings, non-numeric text, NaN, Infinity, zero radius, negative radius, and zero angle are rejected with invalid_input and are not charged. On success the JSON response returns radius and the original angle, the resolved angle_unit, angle_radians and angle_degrees for dual display, centroid_distance as the primary result, and formula as a short string documenting d equals four r sine of half theta over three theta with theta in radians so logs and teaching UIs can cite the algorithm. Example checks: radius five and angle ninety degrees yield a quarter-sector distance along the bisector; radius one and angle pi radians with angle_unit radians yield the semicircle distance four over three pi; angle zero fails naming the degenerate sector. API pricing is the published base of $0.002 per successful item.
Where circular sector centroids show up in real work
Because this capability is a pure parse engine, failed validation does not consume balance and successful runs stay at $0.002 per item.
Integration use cases
Pie-chart and polar UI labels
Turn a radius and central angle into centroid distance for placing labels or balance points on pie wedges in browser or API.
Technical questions
Can I enter the angle in radians?
Yes. Set angle_unit to "radians" (default is "degrees"). The response always returns both angle_radians and angle_degrees.
What about a semicircle or a full circle?
A semicircle (180° or π rad) yields the classic d = 4r/(3π). A full circle (360° or 2π rad) yields d = 0, the disk center, because sin(π) = 0.