Circular segment area
Get the area between a chord and its arc from the radius and central angle: with a radius of 10 at 90 degrees, the segment is the quarter sector minus a triangle of 50. At 180 degrees it becomes a half disk. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 10
- Angle deg
- 90
- Angle rad
- 1.5707963268
- Area
- 28.5398163397
- Sector area
- 78.5398163397
- Triangle area
- 50
A circular segment is the region of a disk cut off by a straight chord: the area between the chord and the arc that subtends a chosen central angle. Given only the circle radius and that central angle in degrees, the segment area follows from a single closed-form identity once the angle is converted to radians. Students meet this figure in homework that asks for the area of a “slice without the triangular bite,” while designers, survey assistants, and makers need the same number when a round plate, tank end, or arch is trimmed by a straight cut.
What a circular segment is and why radius plus central angle fix its area
In Euclidean plane geometry a circle of radius r is fixed once the center and radius are known. Subtracting the triangle from the sector yields the classic segment formula A = (r²/2)(θ − sin θ). At zero degrees the chord collapses and the area is zero; at one hundred eighty degrees the chord is a diameter and the segment is a semicircle of area π r²/2; at three hundred sixty degrees the formula recovers the full disk π r².
Search intent clusters around phrases such as circular segment area, area of a segment of a circle, chord central angle calculator, and sector minus triangle. People want a numeric result they can check against a worksheet key, a CNC cut list, or a CAD note, not only a geometric argument that the figure is determined.
How this calculator validates input and applies the segment formula
The radius must be a positive finite number large enough for stable floating-point evaluation and below a generous upper bound so extreme inputs cannot overflow intermediate products. Degrees are converted with the constant factor π/180 before any trigonometric call. Segment area is then (r²/2) times (angle in radians minus sine of that angle).
Sector area is (r²/2) times the radian measure, triangle area is (r²/2) times sine of the angle, chord length is twice the radius times sine of half the angle, and arc length is radius times the radian measure. Because sine is negative on the open upper half-turn, the same closed form covers major segments without a special branch when the angle exceeds one hundred eighty degrees.
Edge cases are explicit: missing fields, non-numeric strings, zero or negative radius, and angles outside the closed interval all fail before any area is returned.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate segment area while checking homework, sizing a partial tank cross-section, verifying an arch cut-out, or confirming a shop drawing without installing a computer-algebra system. Accepted angles stay inside the closed interval from zero through three hundred sixty degrees; units of length are whatever you already use for the figure, and response lengths and areas share that same unit system. Results are fully deterministic: the same radius and angle always yield the same area, sector, triangle, chord, and arc fields.
There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials.
What you can do with it
Homework and exam circular-segment drills
Students and graders confirm segment area, sector and triangle pieces, chord and arc for classic radius-and-angle data, including rejection when the central angle is outside 0–360°.
FAQ
Which formula does the calculator use?
With θ in radians, segment area is (r²/2)(θ − sin θ). That equals sector area minus the triangular portion formed by the two radii and the chord.
What happens if the angle is not between 0 and 360 degrees?
Angles outside the closed interval [0, 360] are rejected; those calls are not charged.
Is the circular segment area calculator free?
The browser tool runs free on your device.
Does the formula work for major segments above 180°?
Yes. The same closed form uses signed sine, so angles from 180° to 360° return the major segment area without a separate branch.
What else do I get besides the segment area?
You also see the sector area, the triangle that is cut away, the chord length and the arc length, so every step of the sector minus triangle method can be checked.
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/segment-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":10,"angle_deg":90}'const res = await fetch("https://api.kit.forhosting.com/geom/segment-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 10,
"angle_deg": 90
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/segment-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 10,
"angle_deg": 90
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/segment-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":10,"angle_deg":90}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":10,"angle_deg":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/segment-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": 10,
"angle_deg": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.segment_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 capacity accepts a positive radius and a central angle between zero and three hundred sixty degrees inclusive, rejects angles outside that range with a clear invalid_input error, and returns the segment area together with sector area, triangular portion, chord length, and arc length as structured JSON.
What a circular segment is and why radius plus central angle fix its area
A chord at a fixed central angle θ selects a unique pair of arcs—the minor and major paths between the chord endpoints—and the segment belonging to that angle is the area between the chord and the chosen arc. When θ is measured from the two radii that meet the chord endpoints, the sector of angle θ has area (r² θ)/2 with θ in radians, while the isosceles triangle formed by the two radii and the chord has area (r² sin θ)/2. Delivering area, sector_area, triangle_area, chord_length, and arc_length as JSON makes the result easy to paste into notes, diff in automated tests, or feed into a larger pipeline without reimplementing radians conversion and sine in every language you teach or deploy.
How this calculator validates input and applies the segment formula
Send a JSON body with two required numeric fields: radius and angle_deg. The central angle must be between zero and three hundred sixty degrees inclusive; any value strictly below zero or strictly above three hundred sixty fails immediately with invalid_input and a message that names the domain rule, so callers never receive a silent wrap-around or a negative area from a mis-entered angle. Results are rounded to a stable decimal precision so the same input always yields the same JSON.
Practical use, limits, pricing, and delivery channels
Each successful request is one billable item at $0.002; failed validation does not consume balance. Wire the endpoint into segment-area drills, contest answer checkers, tank and pipe segment worksheets, or floating preprocessing steps where you already prefer explicit domain errors over opaque black-box widgets that cannot show why an angle of four hundred degrees is rejected.
Integration use cases
Deterministic fixtures for geometry tests
Pipelines that seed unit tests or interactive segment demos can call a pure circular-segment oracle instead of reimplementing radians conversion and sine in every language.
Tank, arch, and plate cut-out checks
Field or shop notes that record a radius and the central angle of a straight cut recover segment area plus chord and arc length as structured JSON for reports or lightweight CAD helpers.
Technical questions
How do I compute circular segment area with this API?
Send a JSON body with radius and angle_deg (degrees, 0–360) to geom.segment_area. The response returns area, sector_area, triangle_area, chord_length, arc_length, and angle_rad.