ForHosting KIT · Math

Circular segment height calculator

Turn a radius and a chord into the height of the arc above it: a radius of 5 and a chord of 6 give a height of 1, because the chord sits 4 from the center. Handy for templates, arches and fillets. Free in your browser.

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Worked example

If you enter

Radius: 5
Chord: 6

You get

Diameter: 10
Half chord: 3
Height: 1

A circular segment is the region between a chord and the arc it subtends. Builders, machinists, surveyors, and students often know the circle radius and the chord length on a plan or a physical part, but need the sagitta—the perpendicular height from the chord midpoint to the arc—before they can set a template, mill a fillet, or size a liquid meniscus. This circular segment height calculator takes those two lengths, applies the closed-form identity height equals radius minus the square root of radius squared minus half-chord squared, and returns a deterministic height with the echoed inputs, diameter, and half chord.

What segment height means and how the formula works

Drop a perpendicular from the circle center to the chord. That line bisects the chord into two equal half-chords of length c over two, and forms a right triangle with the radius as hypotenuse. Pythagoras therefore gives the distance from the center to the chord as the square root of radius squared minus half-chord squared.

The height of the minor circular segment—the sagitta from the chord up to the nearer arc—is then the radius minus that distance: height equals radius minus square root of radius squared minus open parenthesis chord over two close parenthesis squared. When the chord equals the diameter the distance from center to chord is zero and the height equals the radius, which is a semicircle. When the chord shrinks toward zero the height also shrinks toward zero.

The capability always reports this minor-segment height, which lies between zero and the radius for any valid positive chord up to the diameter. It does not flip to the complementary major-segment height equal to two radius minus the minor height; if you need the larger cap, compute it downstream from the returned height and diameter. Units are arbitrary as long as radius and chord share the same length unit—millimeters, inches, or meters all work because the formula is homogeneous of degree one.

Where the segment height helps

Civil and architectural notebooks convert arch chords and design radii into sagitta so clearance under a vault or crown height of a roadway curve is correct before pour or erection. Classroom geometry and entrance-exam prep need a deterministic oracle that marks impossible chords longer than the diameter as errors rather than silent NaN values a buggy grader might treat as success. Failed validation does not consume balance.

Keep requests small, pin the rounded height in tests, and you get reproducible sagittas for shop floor, LMS plugins, and edge workers without standing up a full computer-algebra stack of your own.

What you can do with it

Shop templates and jigs

From a known arc radius and a measured chord on stock, compute the sagitta so a woodworking or metalworking template matches the intended curve rise.

Arch and roadway clearance

Convert design radius and span chord into segment height for vault clearance, crown height, or arch formwork before construction.

Classroom and exam grading

Check homework and auto-grade geometry items on circular segment height with a deterministic oracle that rejects chords longer than the diameter.

FAQ

How do I calculate circular segment height from radius and chord?

Send positive finite numbers radius and chord (chord ≤ 2×radius).

What happens if the chord is longer than the diameter?

A chord longer than 2×radius cannot lie on that circle, so the geometry is rejected and not billed.

Is the result the minor or major segment height?

Always the minor segment (sagitta from chord to the nearer arc), between 0 and the radius. The complementary major height is 2×radius − height if you need it.

What units does the calculator use?

Any consistent length unit for both radius and chord (mm, in, m, …). The formula is scale-invariant in the sense that height is returned in the same unit.

What if the chord equals the diameter?

Then the chord passes through the center, the segment is a half circle and the height equals the radius. A radius of 5 with a chord of 10 gives a height of 5.

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

POSThttps://api.kit.forhosting.com/geom/segment-height

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-height \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"radius":5,"chord":6}'

const res = await fetch("https://api.kit.forhosting.com/geom/segment-height", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "radius": 5,
    "chord": 6
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/geom/segment-height",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "radius": 5,
    "chord": 6
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/segment-height", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"radius":5,"chord":6}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"radius":5,"chord":6}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/segment-height", 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,
  "chord": 6
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "geom.segment_height",
  "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

Per request$0.002

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

When the chord is longer than the diameter the geometry is impossible on that circle, so the task fails with invalid_input instead of inventing a complex number.

Input contract, response shape, and error behaviour

Radius ten and chord ten yield height approximately one point three three nine seven, matching ten minus the square root of seventy-five. Radius five and chord ten is the semicircle edge case with height five. POST a JSON body with two required finite numbers: radius, a positive circle radius, and chord, a positive chord length that must not exceed twice the radius. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, non-finite values, and non-positive lengths fail with invalid_input naming the problem. On success the body includes the echoed radius and chord, the diameter two times radius, half_chord, the segment height rounded to ten decimal places for stable goldens across runtimes, a formula string restating the closed form, and method equal to segment_height. Example: radius five and chord six yield half_chord three, diameter ten, and height one, because the center-to-chord distance is four and five minus four is one. Pair the response with your own drafting code if you need a sketch; this endpoint returns structured numbers only and does not emit plots or CAD geometry.

Technical questions

What does the circular segment height calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

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