How this instrument works
A cycloid is the path traced by a single point fixed to the rim of a circle as that circle rolls, without slipping, along a straight line. Track the valve stem on a bicycle wheel through one full turn and you have drawn one arch: it lifts off the ground, loops up and over the top, then lands again exactly one circle-turn away. This is not the rolling circle's own circumference or area — those are the tidy 2πr and πr² of the circle itself — but two different figures describing the humped shape the rim point sweeps out overhead.
Both results come out surprisingly clean given how contorted the curve looks. Parametrise the point's position as the circle turns through angle θ, and its speed along the path works out to 2r·sin(θ⁄2); integrate that speed over one full revolution, θ from 0 to 2π, and it collapses to exactly 8r — no π survives the arc-length calculation at all. The area underneath does keep a π: integrating the height against the horizontal sweep gives 3πr², precisely three times the area of the generating circle. Roberval and Torricelli proved that 3-to-1 area ratio independently in the 1630s and 1640s, and Christopher Wren settled the arc length in 1658, answering a rectification challenge Blaise Pascal had posed that year.
Scale behaves differently for the two outputs. Arc length grows in direct proportion to the radius — double r and the arch is exactly twice as long, since 8r is a straight line through the origin — while area grows with the square of the radius, because 3πr² quadruples whenever the linear size doubles, the same as any flat figure. At the extreme, a rolling circle with zero radius traces no arch at all: both figures collapse to zero, the flat line the point never leaves.
- Enter the rolling circle's size into Rolling circle radius — any positive length unit works, as long as you read the results in that same unit.
- Read Arc length (one arch) for the total distance the rim point travels over one full revolution of the circle.
- Read Area under one arch for the region enclosed between the arch and the flat line the circle rolls along.
- Set Rolling circle radius to zero to see both outputs collapse to the degenerate case: a point that traces no curve at all.
Worked example — a radius-2 rolling circle
Roll a circle of radius 2 along a flat line for one full turn. Rolling circle radius is 2, so Arc length (one arch) comes out to L = 8 × 2 = 16 exactly, with no rounding, since 8r is a plain multiple of an integer input. That is longer than the straight-line distance the circle itself covers in one revolution, 2π × 2 ≈ 12.566, because the arch bows up and over the top of the circle before it comes back down to the line.
Area under one arch works out to A = 3π × 2² = 12π ≈ 37.699112 — three times the area of the radius-2 rolling circle itself, π × 2² = 4π ≈ 12.566. Both figures match the sheet's own output exactly: Arc length (one arch) reads 16.0 and Area under one arch reads 37.69911184307752, the same closed-form 12π carried to full precision.
Questions
What is a cycloid, exactly?
It is the curve traced by one fixed point on the rim of a circle as the circle rolls, without slipping, along a straight line. One full turn of the circle draws one arch of the cycloid — the shape this sheet measures — before the pattern repeats indefinitely along the line.
Why is the arc length exactly 8 times the radius, with no π?
Because integrating the tracing point's speed over one full revolution happens to cancel every factor of π in the working, leaving the plain multiple 8r. Christopher Wren proved this in 1658, answering a rectification challenge that Blaise Pascal had posed to European mathematicians that same year.
How is the area under a cycloid arch related to the rolling circle?
It is exactly three times the rolling circle's own area: 3πr² against the circle's πr². Roberval and Torricelli both proved this ratio independently in the 1630s and 1640s, and it remains one of the most quoted results in the history of curve quadrature.
Is a cycloid's arc length the same as a circle's circumference?
No — they are different curves with different formulas. A circle of radius r has circumference 2πr, but one arch of the cycloid traced by a point on that same circle's rim measures 8r, nearly 27 percent longer than the circle's own distance around.
What happens to the formulas beyond one arch?
They describe a single arch only, covering one full revolution of the rolling circle, θ from 0 to 2π. Past that point the cycloid simply repeats an identical arch after every revolution, so a stretch covering n arches has length 8rn and area 3πr²n.
Where does the name 'cycloid' come from?
From the Greek kyklos, meaning circle, since the whole curve is generated by a circle's rolling motion. Galileo is credited with naming and popularising it in the early 1600s, though he never found its exact area — he tried weighing cut-out shapes against circles instead.