Circular segment chord calculator
Get the straight base of a circular segment from the radius and central angle: a radius of 10 at 90 degrees gives a chord of 10√2, together with the half-chord and the height up to the 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 deg
- 90
- Angle rad
- 1.5707963268
- Chord length
- 14.1421356237
- Half chord
- 7.0710678119
- Sagitta
- 2.9289321881
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. Before you can size that figure, lay out a cut, or check a worksheet, you often need the length of the chord itself—the straight base of the segment. Given only the circle radius and the central angle in degrees, the chord length follows from a single closed-form identity once the angle is converted to radians: twice the radius times the sine of half the angle. Students meet this length in homework that asks for the base of a segment or the side opposite a central angle, while designers, survey assistants, and makers need the same number when a round plate, tank end, arch, or pipe is trimmed by a straight cut.
What a circular-segment chord is and why radius plus central angle fix its length
In Euclidean plane geometry a circle of radius r is fixed once the center and radius are known. At zero degrees the chord collapses and the length is zero; at one hundred eighty degrees the chord is a diameter and L equals 2r; at ninety degrees you recover the familiar r√2 result that appears in countless textbook drills. Search intent clusters around phrases such as circular segment chord calculator, chord length of a segment, chord from radius and central angle, and base of a circular segment.
People want a numeric length 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 chord 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. Chord length is then twice the radius times the sine of half the angle in radians.
Half-chord is simply that length divided by two. Sagitta for the minor arc is computed by folding the absolute angle into a half-turn and applying r times one minus cosine of half that folded angle, which gives the height from the chord to the nearer arc—useful when a shop drawing records sagitta instead of angle. Edge cases are explicit: missing fields, non-numeric strings, zero or negative radius, and negative angles all fail before any chord length is returned.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate segment chord while checking homework, sizing a straight cut across a round plate, verifying an arch opening, or confirming a shop drawing without installing a computer-algebra system. Accepted angles are non-negative degrees; units of length are whatever you already use for the figure, and response lengths share that same unit system. 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.
If you need segment area instead of chord length, pair this tool with the companion circular-segment area capability that uses the same radius-and-angle framing.
What you can do with it
Homework and exam circular-segment chord drills
Students and graders confirm chord length, half-chord, and sagitta for classic radius-and-angle data, including rejection when the central angle is negative.
FAQ
Which formula does the calculator use?
With θ in radians, chord length is 2 r sin(θ/2).
What happens if the central angle is negative?
Negative angles are rejected with a message that the central angle must not be negative; those calls are not charged.
Is the circular segment chord calculator free?
The browser tool runs free on your device.
How is this different from a general chord-length calculator?
This capacity is framed for circular-segment landings: radius and central angle only, with an explicit negative-angle error. A general chord tool may also accept distance from center or other modes; here the SEO intent is segment chord from angle.
What do the half-chord and the height tell me?
The half-chord is half the base, measured from the middle of the chord to one end. The height runs from the chord to the nearer arc, which helps when a drawing records that height instead of the angle.
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-chord \
-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-chord", {
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-chord",
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-chord", 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-chord", 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_chord",
"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 non-negative central angle, rejects a negative angle with a clear invalid_input error, and returns the chord length together with half-chord, minor-arc sagitta, and the radian measure as structured JSON.
What a circular-segment chord is and why radius plus central angle fix its length
Two radii that meet the endpoints of a chord form a central angle θ; the chord is the straight side of that isosceles triangle, and its length is completely determined by r and θ. With θ in radians the half-angle formula is L = 2 r sin(θ/2). Delivering chord_length, half_chord, sagitta, and angle_rad 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. This landing is aimed at segment-first queries—the chord as the base of a segment—rather than a general multi-mode chord tool that also takes distance from center.
How this calculator validates input and applies the chord formula
Send a JSON body with two required numeric fields: radius and angle_deg. The central angle must be non-negative; any strictly negative value fails immediately with invalid_input and a message that names the domain rule, so callers never receive a silent absolute-value wrap or a meaningless length 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. Results are fully deterministic: the same radius and angle always yield the same chord_length, half_chord, sagitta, and angle_rad fields. Wire the endpoint into segment-chord drills, contest answer checkers, tank and pipe cut worksheets, or floating preprocessing steps where you already prefer explicit domain errors over opaque black-box widgets that cannot show why a negative central angle is rejected.
Integration use cases
Deterministic fixtures for geometry tests
Pipelines that seed unit tests or interactive segment demos can call a pure segment-chord oracle instead of reimplementing radians conversion and sine in every language.
Plate, arch, and pipe cut-out checks
Field or shop notes that record a radius and the central angle of a straight cut recover chord length plus sagitta as structured JSON for reports or lightweight CAD helpers.
Technical questions
How do I compute circular segment chord length with this API?
Send a JSON body with radius and angle_deg (degrees, non-negative) to geom.segment_chord. The response returns chord_length, half_chord, sagitta, and angle_rad.