Cycloid arc length and area calculator
Give the radius of the rolling circle and get one full cycloid arch: a radius of 1 gives an arch length of 8 and an area of 3π, three times the area of the circle itself. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 1
- Arch length
- 8
- Area
- 9.4247779608
- Formula arch length
- L = 8 · r
- Formula area
- A = 3 · π · r²
A cycloid is the curve traced by a point fixed on the rim of a circle that rolls without slipping along a straight line. One full arch—from cusp to cusp—has two classic closed forms that appear in calculus courses, brachistochrone history, and mechanical design: the arch length is exactly eight times the generating radius, and the area under that arch is three times pi times the radius squared. Hand derivation of those identities needs a parametric integral and a careful choice of limits; numerical quadrature is unnecessary once the closed forms are known. Zero radius is treated as a degenerate point cycloid with length and area zero so callers do not need a special branch.
What a cycloid arch is and how the closed forms work
Place a circle of radius r on the x-axis and mark a point on its circumference. As the circle rolls to the right without slipping, that point traces a cycloid with parametric equations x equals r times open parenthesis t minus sine of t close parenthesis and y equals r times open parenthesis one minus cosine of t close parenthesis.
One arch corresponds to the parameter interval from zero to two pi, starting and ending at a cusp on the base line and peaking at height two r in the middle. The speed of the parametric path is two r times the absolute value of sine of t over two, so integrating the speed from zero to two pi yields the arc length eight r—four times the diameter of the generating circle.
Integrating the product of y and dx over the same interval yields the area three pi r squared, which is three times the area of the generating disk. Those two identities are the entire contract of this calculator. It does not return the cusp-to-cusp base width two pi r as a separate field, the height two r, a multi-arch concatenation, a prolate or curtate trochoid, or an inverted cycloid for tautochrone problems.
Keeping the output to radius, arch length, area, and formula strings makes the tool ideal as a fast check beside a textbook problem, a gear-path dimension, or a one-line step in a larger geometry pipeline that already chose the standard rolling-circle model.
Where a cycloid arch calculator earns its keep
Geometry and calculus homework often asks for the length of one cycloid arch or the area under it after students have already written the parametric integrals. Learners sometimes replace the integral by the generating circumference or by the base width two pi r, which underestimates the true arch length eight r; a deterministic oracle that always applies the closed form catches those shortcuts before a tutor spends a session on half-angle substitutions.
Mechanical and cam-design notebooks that freeze a rolling radius and need the developed length of a cycloidal path or the area of a cycloidal lobe want a closed-form number without importing quadrature libraries. Architecture and landscape workflows that lay cycloidal arches, ramps, or decorative profiles convert a measured generating radius into total curve length and enclosed area for stock cut lists without opening a full CAD suite.
Exam-prep generators and LMS auto-graders need a tiny closed-form checker that rejects negative radii as invalid rather than silently returning a signed length that a buggy grader might accept.
What you can do with it
Cam paths and material estimates
Convert a frozen rolling radius into total cycloidal path length and enclosed lobe area for stock, cam, or profile estimates without numerical integrators.
LMS and exam auto-grading
Mark cycloid arch items with a deterministic oracle that rejects negative radii as invalid and never invents a signed length for an impossible generator.
FAQ
What happens if the radius is negative?
A generating-circle radius cannot be less than zero; those failures are not billed.
Does this cover prolate, curtate, or multi-arch cycloids?
No. The calculator is limited to one standard (cuspidal) arch from a circle rolling on a straight line. Other trochoids need a different tool.
How long is one arch of a cycloid?
Exactly eight times the radius of the rolling circle, which is four times its diameter. A circle of radius 1 traces an arch 8 units long from one cusp to the next.
What is the area under a cycloid arch?
It is three times pi times the radius squared, or three times the area of the rolling circle. The region is the one between a single arch and the straight line the circle rolls on.
Why is the arch longer than the base it spans?
The base from cusp to cusp is 2πr, the circle's circumference, while the arch itself measures 8r. Using the circumference or the base width in place of the arch length underestimates it, a common slip in homework.
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/cycloid \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":1}'const res = await fetch("https://api.kit.forhosting.com/geom/cycloid", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/cycloid",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/cycloid", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/cycloid", 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": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.cycloid",
"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 cycloid arc length and area calculator accepts the generating-circle radius, rejects a negative value with a clear invalid_input error, and returns the positive arch length and enclosed area together with short formula strings.
What a cycloid arch is and how the closed forms work
When r is zero both results collapse to zero; when r is negative the request fails rather than inventing a meaningless signed length.
Input contract, response shape, and error behaviour
Example: radius equals one yields arch length eight and area about nine point four two four seven seven seven nine six zero seven seven, matching the classic unit-radius cycloid. Units are caller-defined: if you send metres for r you get metres of arch length and square metres of area; if you send inches you get inches and square inches. The calculator does not convert units, draw the curve, solve the brachistochrone variational problem, or switch to a hypocycloid or epicycloid family. Pair the numeric results with your own unit labels and drawings. Keep requests small, pin the rounded fields in tests, and you get reproducible one-field answers for homework, LMS plugins, and edge workers without standing up a numerical integrator. POST a JSON body with one required finite number: radius, the generating-circle radius r of the standard cycloid. The field may arrive as a JSON number or as a numeric string; empty strings, missing fields, Infinity, and NaN fail with invalid_input naming the problem. Aliases r, a, and R map to radius so clients that already use a short letter for the rolling circle do not need a rename layer. On success the body includes the echoed radius rounded to twelve decimal places for stable goldens; arch_length equal to eight times r with the same rounding; area equal to three times pi times r squared with the same rounding; formula_arch_length and formula_area restating L equals eight r and A equals three pi r squared; and method equal to the stable token cycloid_arch_length_area.
Where a cycloid arch calculator earns its keep
Engineering notebooks and spreadsheet replacements can call this endpoint as a one-field step when a larger script already holds radii for several figures and only the pure cycloid arch case needs a dedicated, SEO-clear name distinct from circular sector arc length, parabola arc length, or astroid perimeter suites. In all of those settings the value is not exotic mathematics: it is a stable contract, clear errors, and a search intent that matches how people actually query for cycloid arch length and area from a generating radius at $0.002 per successful API call.
Integration use cases
Classroom and homework checks
Students enter the generating radius and verify L=8r and A=3πr² for one arch, catching circumference or base-width approximations that undercount the true curve.
Technical questions
How do I compute the arc length and area of a cycloid arch?
Send a finite non-negative radius r. The API returns arch_length = 8r and area = 3πr² for one standard arch from cusp to cusp.
Is radius = 0 allowed?
Yes. Zero yields a degenerate point cycloid with arch_length 0 and area 0, which is a valid success rather than an error.
What does the cycloid calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.