Segment area calculator
A circle of radius 10 cut by a chord at a 90° central angle leaves a segment of 28.539816339745 square units. Enter the radius and the central angle in degrees to get the area between the chord and its arc. 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
- 90
- Angle rad
- 1.5707963268
- Area
- 28.5398163397
A circular segment is the leftover region between a chord and the arc that chord subtends: the cap that remains after a straight cut slices a disk. Students meet the figure when a worksheet asks for the area of a slice after the triangular bite is removed from the sector. Inspectors, shop machinists, and tank designers need the same number when a round plate, vault, or pipe is trimmed by a straight cut. This calculator accepts a positive radius and a central angle in degrees, converts the angle to radians, and returns the segment area from the identity one-half r squared times theta minus sine of theta.
What a segment area is and why radius plus central angle fix it
In plane geometry a circle is fixed once its center and a positive radius are known. The sector area is one-half r squared times the central angle in radians. The triangular portion is one-half r squared times the sine of that same radian measure.
Subtracting the triangle from the sector leaves the classic segment-area identity: one-half r squared times theta minus sine of theta. At zero degrees the chord collapses to a point and the area is zero. At one hundred eighty degrees the chord is a diameter and the segment is a semicircle of area one-half pi r squared.
At three hundred sixty degrees the same formula recovers the full disk pi r squared, because sine of two pi is zero. Searchers who type segment area, area of a segment of a circle, or segment area calculator want that numeric leftover, not a lecture that only argues the figure is determined.
Delivering the echoed radius, the angle in degrees and radians, the computed area, and a short formula string makes the result easy to paste into notes, pin in automated tests, or feed a larger pipeline without reimplementing the radians conversion and the sine call in every language you teach or deploy. The identity is deterministic and constant time, so the same two numbers always produce the same area.
What you enter, the formula and what you get
The radius must be a strictly positive finite number: zero and any negative length are rejected by name, because they do not describe a circle you can draw. The angle must lie in the closed interval from zero through three hundred sixty degrees; values outside that interval fail immediately instead of wrapping around or returning a negative leftover. Degrees are converted with the constant factor pi over one hundred eighty before any trigonometric call.
Segment area is then one-half the square of the radius times the radian measure minus the sine of that radian measure. Because sine is an odd function around pi, the same closed form covers major segments above one hundred eighty degrees without a special branch. Example checks: radius ten and angle ninety yield a quarter-turn leftover whose area is fifty times pi over two minus one; radius five and angle one hundred eighty yield a semicircle of twelve and a half times pi.
Where a segment-area calculator earns its keep
Classrooms and tutoring platforms use a segment area calculator so learners can check the leftover cap for arbitrary radius and central-angle pairs without mixing up degrees and radians by hand. Exam-prep generators need a deterministic oracle that marks a non-positive radius as invalid instead of silently returning zero, which a buggy grader might treat as success.
Shop, tank, and civil notes that record a radius and the opening angle of a straight cut recover a stable area for material takeoff, fill volume of a partial circular cross-section, or a CNC cut list. Failed validation does not consume balance. Keep requests small, pin the rounded area in tests, and you get reproducible leftovers for homework, LMS plugins, and edge workers without standing up a computer-algebra system.
What you can do with it
Homework and exam segment-area drills
Students and graders confirm the leftover cap for classic radius-and-angle data, including rejection when the radius is not a positive number.
FAQ
Which formula does the segment area calculator use?
With the central angle converted to radians, segment area is (r²/2)(θ − sin θ). That equals the circular sector minus the isosceles triangle formed by the two radii and the chord.
What happens if the radius is not a positive number?
Zero, a negative length, a missing field, or a non-finite value is not a drawable circle; those calls are not billed.
Is the segment area calculator free?
The browser tool runs free on your device.
Does the formula work for major segments above 180 degrees?
Yes. The same closed form uses signed sine, so angles from 180° to 360° return the major leftover without a separate branch. An angle outside 0–360 is rejected.
What area do 0°, 180° and 360° give?
At 0° the chord shrinks to a point and the area is zero. At 180° the chord is a diameter and the segment is a semicircle, half of pi times the radius squared; at 360° you get the whole disk.
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/math/segment-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":10,"angle":90}'const res = await fetch("https://api.kit.forhosting.com/math/segment-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 10,
"angle": 90
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/segment-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 10,
"angle": 90
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/segment-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":10,"angle":90}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":10,"angle":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/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": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.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
Given only the circle radius and the central angle formed by the two radii that meet the chord endpoints, the segment area is a single closed-form evaluation. A radius that is missing, not finite, or not strictly positive fails with invalid_input instead of a silent nonsense area. The same pure function powers the free browser tool and the prepaid API path.
What a segment area is and why radius plus central angle fix it
A chord at a chosen central angle selects a unique pair of arcs between its endpoints, and the circular segment belonging to that angle is the area between the chord and the chosen arc. Draw the two radii to the chord endpoints: those radii and the chord form an isosceles triangle, while the same radii and the arc form a circular sector.
Input contract, formula, and what the response contains
Send a JSON body with two required numbers: radius, the positive circle radius in any consistent length unit, and angle, the central angle of the segment in degrees. Missing fields, empty strings, non-numeric text, NaN, and Infinity fail with invalid_input naming the field and are not billed. A successful body echoes radius and angle, includes angle_rad for transparency, returns area as the primary value, and restates the formula so logs and teaching interfaces can document the algorithm. Pair the JSON with your own sketch if you need a drawing; this endpoint computes the area only.
Where a segment-area calculator earns its keep
If you later need chord length, sagitta height, sector perimeter, or an elliptical segment, treat those as neighbouring search intents rather than overloading this landing: people who type segment area want the circular leftover from radius and central angle, returned as structured JSON they can trust.
Integration use cases
Deterministic fixtures for geometry tests
Pipelines that seed unit tests or interactive segment demos can call a pure segment-area oracle instead of reimplementing radians conversion and sine in every language.
Tank, plate, and arch cut-out checks
Field or shop notes that record a radius and the central angle of a straight cut recover the circular leftover as structured JSON for reports or lightweight CAD helpers.
Technical questions
How do I compute segment area from radius and central angle?
Send a JSON body with radius (a positive finite number) and angle (degrees, 0 through 360). The response returns area from (r²/2)(θ − sin θ) with θ in radians, plus the echoed inputs and the formula string.