Hong Life CalculatorsUnderstanding the cycloid calculator — concept and principle
This calculator describes the cycloid, the curve traced by a point on the rim of a rolling circle. From the circle's radius it gives the length, width, height and area of one arch, and for a chosen turning angle it gives the point's position and the distance it has travelled. Use it for calculus problems, gear and wheel design ideas, or curiosity about the "fastest descent" curve.
How it is calculated
Point: x = r(t − sin t), y = r(1 − cos t), with t in radians
One arch: width 2πr, height 2r
Area under one arch = 3πr²
Arc length from 0 to t (within one arch) = 4r(1 − cos(t/2)); one full arch = 8r
- Lengths are in the unit of r, areas in its square.
The principle
As the circle rolls without slipping, its centre moves forward by rt while the point turns around the centre by angle t. Adding those two motions gives the parametric equations. The point touches the ground at the start of each turn, where it is momentarily at rest, and is highest after half a turn, at height 2r. Integrating the speed √(x′² + y′²) = 2r sin(t/2) gives the arc length, and integrating y dx gives the area, three times that of the circle itself. The cycloid is also the curve of fastest descent under gravity (the brachistochrone).
Worked example
Default inputs: r = 35 cm, t = 90°.
- One arch: length 8 × 35 = 280 cm; width 2π × 35 = 219.911 cm; height 2 × 35 = 70 cm.
- Area under one arch = 3π × 35² = 11,545.4 cm².
- At t = 90° = π/2: x = 35(π/2 − 1) = 19.9779 cm; y = 35(1 − 0) = 35 cm.
- Path travelled so far = 4 × 35 × (1 − cos 45°) = 41.0051 cm.
| Radius of the rolling circle r (cm) | Length of one arch (8r) |
|---|---|
| 8.75 | 70 |
| 17.5 | 140 |
| 26.2 | 210 |
| 35 | 280 |
| 43.8 | 350 |
| 52.5 | 420 |
| 70 | 560 |
How to read the result
After a quarter turn the point has risen to the height of the centre but has moved forward only about 20 cm, while the wheel's centre has moved 55 cm: the rim point lags behind at first and then overtakes. Over a full turn the point travels 280 cm along its curved path while the wheel rolls 219.9 cm forward.
Common mistakes
- Using degrees inside sin and cos without converting to radians.
- Confusing the arch width 2πr with the arch length 8r.
- Applying the arc-length formula beyond one arch.
- Measuring r as a diameter.
Frequently asked questions
How long is one arch of a cycloid?
Exactly 8 times the radius of the rolling circle.
What is the area under a cycloid arch?
3πr², three times the area of the circle.
Why is the cycloid famous?
It is the curve of fastest descent between two points under gravity, and of equal descent time from any start (tautochrone).
Sources: OpenStax, Calculus Volume 2 (2016), §7.2 Calculus of Parametric Curves