Cycloid
Enter the radius of a rolling wheel and get the length of one cycloid arch, the area beneath it or any point on the curve: a radius of 2 gives an arch 16 long, always exactly eight times the radius. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 2
- Calculation mode
- Arc length
- Arc length
- 16
The cycloid is the curve traced by a point on the rim of a wheel as it rolls along a straight line, and it shows up everywhere from gear teeth and pendulum clocks to the fastest-descent problem in physics. This capability takes the wheel radius and returns the quantity you actually need: the length of one full arch (always exactly 8r), the area under one arch (3πr²), or the exact coordinates of the curve at any roll angle.
What a cycloid is and why the radius is enough
Mark a single point on the rim of a bicycle wheel and watch it as the wheel rolls without slipping along the ground: the point rises, reaches its peak when the wheel has turned half a revolution, and comes back down to touch the ground after one full turn. The path it draws is a cycloid.
The entire shape is fixed by a single number — the wheel radius r — because the curve is defined parametrically as x(t) = r·(t − sin t) and y(t) = r·(1 − cos t), where t is the roll angle in radians. One arch corresponds to t running from 0 to 2π. No other measurement is needed: scale the wheel and the whole curve scales with it, which is why this capability asks for nothing more than a positive radius and the mode you want.
Arc length and area: two closed forms worth knowing
Two facts about the cycloid arch are famous precisely because they are so clean. The length of one arch is exactly 8r — four diameters of the wheel — a result found geometrically by Christopher Wren in 1658, long before calculus made it routine. The area under one arch is exactly 3πr², which is three times the area of the generating circle; Galileo guessed the ratio empirically by weighing cut-outs, and Torricelli later proved it.
This capability returns both values directly from those formulas: send a radius of 2 and you get an arc length of 16 and an area of 3π·4 ≈ 21.205750. There is no numerical integration and no approximation step, so repeated calls with the same radius always produce byte-identical results.
If you are comparing against a textbook, keep in mind that some authors define the arch from a wheel rolling along the top of a line and flip the vertical axis; the length and area are unaffected, because both depend only on the radius and not on the orientation you choose for the drawing.
Coordinates at a roll angle, and how the modes fit together
Sometimes you do not want a summary of the whole arch but a specific point on it — to place a keyframe in an animation, to sample the profile of a gear flank, or to plot the curve. The coordinates mode evaluates the parametric equations at the roll angle t you provide and returns x and y. Results are rounded to six decimal places for stability.
What you can do with it
Animate a rolling wheel
Sample coordinates at successive roll angles to keyframe the rim point of a wheel in a physics-accurate animation.
Size a cycloidal gear profile
Get the arch length and area from the pitch radius when drafting cycloidal gear teeth or clock escapements.
Teach the brachistochrone
Show students the exact arc length 8r and area 3πr² alongside plotted coordinates of the fastest-descent curve.
FAQ
What input does it need?
The coordinates mode also needs the roll angle t.
What units are the results in?
Whatever unit you give the radius in. If the radius is in centimetres, the arc length is in centimetres and the area in square centimetres.
Which values of t make sense for coordinates?
Any finite number works, since the equations are periodic in structure; the first arch corresponds to t between 0 and 2π (about 6.283185).
Are the results exact?
Arc length and area use the exact closed forms 8r and 3πr²; coordinates come straight from x = r(t − sin t), y = r(1 − cos t). All numeric output is rounded to six decimal places for stable comparisons.
Where is the highest point of a cycloid arch?
Halfway through the roll, when the wheel has turned half a revolution and the angle is π. There the point sits at a height of twice the radius, π times the radius along the line.
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/cycloid \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":2}'const res = await fetch("https://api.kit.forhosting.com/math/cycloid", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/cycloid",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/cycloid", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/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": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.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
Every result is computed from closed-form formulas, so it is deterministic, instant, and identical on every call.
Coordinates at a roll angle, and how the modes fit together
At t = π the wheel has rolled half a turn and the point sits at the top of the arch: x = πr, y = 2r. At t = 2π it is back at ground level, one full arch-width of 2πr down the line. The same logic runs free in your browser on this page and behind the API at $0.002 per call, so you can prototype by hand and automate later without the numbers ever changing.
Technical questions
What does it cost?
$0.002 per API request. It is also free to run in your browser on this page.
What happens if the radius is zero or negative?
The request is rejected with an invalid_input error and is not charged. Only positive finite radii are accepted.