How this calculator works
A cycloid is traced by a point on the rim of a circle as the circle rolls without slipping along a straight line. One complete arch is produced by exactly one full turn of the circle, so its base length is the circumference of the rolling circle. The remarkable result is that the arch area is three times the area of the circle and the arc length is eight times the radius, both independent of how far the circle has rolled. The perimeter of the arch is the curved arc plus its straight base.
Formula
For a rolling circle of radius r: area of one arch = 3πr²; arc length = 8r; hump height = 2r; hump length = 2πr; perimeter = 8r + 2πr.
Worked example
A rolling circle of radius 1 m produces an arch with area 3π ≈ 9.4248 m², arc length 8 m, height 2 m, base length 2π ≈ 6.2832 m and perimeter 8 + 2π ≈ 14.2832 m.
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Frequently asked questions
What is a hump?
A hump is one complete arch of the cycloid, the curve from one point where the rolling point touches the line to the next. Its base is one circumference of the rolling circle.
Why is the area exactly three circles?
The result follows from integrating the height of the rolling point over one turn. It was proved by Roberval in the seventeenth century and does not depend on the radius.
Does the perimeter include the base?
Yes. The arch outline is the curved arc plus the straight base between its two ends. If you only need the curved part, use the arc length result.
Can a cycloid be longer than eight radii?
Only if more than one arch is measured. A single arch has arc length exactly 8r; consecutive arches simply repeat that value.