# Cycloid Calculator

URL: https://speedycalculator.com/calculators/math/cycloid/
Last updated: 2026-10-30

> The cycloid calculator finds the area under one arch, the arc length, the arch height and width, and the perimeter of the region under a cycloid traced by a point on a rolling circle of radius r.

## Inputs

- Radius of the rolling circle r (cm, m, mm, in, ft, yd, km, mi)

## Results

- Area under one arch
- Arc length of one arch
- Arch height
- Arch width
- Perimeter of the region under one arch

## Formula

```
Area = 3πr²,  arc length = 8r,  height = 2r,  width = 2πr
x = r(t − sin t),  y = r(1 − cos t)
```

## Explanation

A cycloid is the path of a point on the rim of a rolling wheel. It is the brachistochrone – the curve of fastest descent – and the tautochrone, along which a ball reaches the bottom in the same time from any starting point. Galileo, Pascal and Bernoulli all studied it.

### How to use the Cycloid Calculator

- **Radius of the rolling circle r**

Results update as you type, and everything is calculated in your browser.

### How it works

Enter the radius of the rolling circle. One arch is traced by one full turn of the wheel. The area under one arch is exactly three times the area of the wheel.

### What each result means

- **Area under one arch** = 3×pi×[Radius of the rolling circle r]^2
- **Arc length of one arch** = 8×[Radius of the rolling circle r]
- **Arch height** = 2×[Radius of the rolling circle r]
- **Arch width** = 2×pi×[Radius of the rolling circle r]
- **Perimeter of the region under one arch** = 8×[Radius of the rolling circle r] + 2×pi×[Radius of the rolling circle r]

### Example values

How the main result changes when **Radius of the rolling circle r** changes and everything else stays at the example values above:

Radius of the rolling circle r (cm)Area under one arch (cm²)2.558.90493.75132.53595235.61946.25368.15547.5530.143810942.4778

## Worked example

Inputs — Radius of the rolling circle r: 5 cm.

Results — **Area under one arch = 235.619449 cm²**; Arc length of one arch = 40 cm; Arch height = 10 cm; Arch width = 31.415927 cm; Perimeter of the region under one arch = 71.415927 cm.

## Key terms

- **Cycloid**: The curve traced by a point on a rolling circle.

## FAQ

**What is the area under a cycloid?**
Exactly 3πr², three times the wheel's area.

**Why is the arc length 8r?**
Integrating the speed of the point over one turn gives 8r.

**What is the brachistochrone problem?**
Finding the fastest slide between two points – the answer is a cycloid.

**Do all points of a wheel trace cycloids?**
Points on the rim do; inner points trace curtate cycloids.


## Sources

- [Wolfram MathWorld – Cycloid](https://mathworld.wolfram.com/Cycloid.html)
- [OpenStax – Calculus Volume 2](https://openstax.org/details/books/calculus-volume-2)