How this instrument works
Roll a circle of radius r once around the outside of a fixed circle of radius R, keeping contact without slipping, and count how many times the rolling circle spins on its own axis. The answer is 1 + R ⁄ r — not the R ⁄ r that a glance at the two circumferences suggests. Two identical coins already make the point: with R equal to r the rolling coin turns twice, not once, in a single lap, which is why this result keeps its name as a paradox rather than a mere formula.
The rolling contact alone accounts for exactly R ⁄ r of that spin: it is the arc length traced along the fixed circle's rim, 2πR, divided by the rolling circle's own circumference, 2πr — the number of times its edge must turn over to lay down that much arc without skidding. The missing turn comes from revolution, not rotation. As the rolling circle's center sweeps one full loop of its own, a path of radius R + r around the fixed circle's center, the whole rolling circle is carried through one extra full turn of orientation, the way a compass needle taped flat to a merry-go-round horse still ends up pointing every direction once per ride even if nobody twists it by hand. Add that carried turn to the rolling turns and the +1 falls out.
The same bookkeeping explains why Earth completes about 366.242 rotations relative to the distant stars in a year but only 365.242 sunrises: the extra spin is exactly one turn, contributed by Earth's own orbit around the Sun, not by the planet spinning any faster. Flip the setup and roll the same circle around the inside of a fixed ring instead, and the sign flips too — the count becomes R ⁄ r − 1, one turn fewer than the contact-only guess, because revolution and rotation now work against each other. Set R = 2r in that inside case and the rolling circle turns exactly once per lap, the special case behind the Tusi couple, a device 13th-century astronomers used to trace a straight line from pure circular motion.
- Enter the radius of the circle that stays put into Fixed circle radius (R) — any unit, so long as the next field matches it.
- Enter the radius of the circle that rolls around it into Rolling circle radius (r), using that same unit.
- Read Rotations completed (rolling around the outside) for the exact number of full spins in one complete lap.
- Set both radii equal first to see the paradox at its sharpest: two matched circles always give 2 rotations, never 1.
Worked example — two matched coins
Take two identical coins of radius 1 (one inch, one centimetre — the unit cancels either way). Fix one flat on the table and roll the second around its rim, staying on the outside, back to its starting point, without letting it slip. Fixed circle radius R = 1 and Rolling circle radius r = 1 feed the formula directly: rotations = 1 + R ⁄ r = 1 + 1 ⁄ 1 = 1 + 1 = 2. Rotations completed reads 2.0 — the rolling coin spins twice before it returns to where it began, despite travelling a path only as long as one circle its own size.
Most people guess 1, reasoning that a coin circling a track exactly its own size should turn over exactly once — the logic that does work for a coin rolled along a straight ruler one circumference long. The extra spin costs no extra distance; it comes entirely from the coin's center travelling a circle instead of a straight line while it rolls. Mark the coin's face, roll it around a second coin of the same size by hand, and count the mark returning to the top — it happens twice, confirming the 2.0 this sheet reports.
Questions
Why is the rotation count 1 + R ⁄ r instead of just R ⁄ r?
R ⁄ r counts only the spin caused by rolling contact — the arc length laid down along the fixed circle's rim divided by the rolling circle's own circumference. The extra +1 comes from revolution: the rolling circle's center travels one full loop of its own around the fixed circle's center, and that loop alone reorients the rolling circle by one additional full turn, no matter how large or small either radius is. Rolling and revolving add together here, which is why the terms sum instead of cancelling.
What happens when the two circles are exactly the same size?
Set R equal to r and the formula reduces to 1 + 1 = 2: any two matched circles, rolled one around the other without slipping, produce exactly two rotations per lap, regardless of how large or small the matched pair is. This is the paradox at its sharpest, because a circumference-only argument — same size, so one lap should equal one spin — gives the wrong answer of 1.
Does rolling on the inside of a ring give the same formula?
No — rolling inside a fixed ring flips the sign to R ⁄ r − 1, one turn fewer than the contact-only count instead of one more. The case R = 2r gives R ⁄ r − 1 = 1 exactly: a circle rolling inside a ring twice its size spins once per lap, and a point fixed to its rim traces a straight line back and forth across the ring's diameter, a construction known since the 13th-century astronomer Nasir al-Din al-Tusi.
Is this the same effect behind the gap between a solar day and a sidereal day?
Yes. Earth's own spin plus its yearly orbit around the Sun means Earth completes one more rotation relative to the distant stars over a year than it does relative to the Sun — about 366.242 sidereal days against 365.242 solar days. That spare rotation is contributed purely by orbital revolution, exactly the +1 term in this formula, not by Earth spinning any faster on its axis.
How does the rotation count behave when one radius is much bigger than the other?
As r shrinks toward zero relative to R, R ⁄ r grows without bound, so a tiny rolling circle spins enormously many times circling one large fixed circle, and the +1 becomes negligible beside it. Run it the other way, with r far larger than R, and the count barely exceeds 1, since R ⁄ r shrinks toward zero and the single revolution accounts for almost the whole answer.
What is the most common mistake people make with this formula?
Treating R ⁄ r as the complete answer and forgetting the +1 contributed by revolution — the same slip that makes a circumference-matching argument look right when it is not. It shows up anywhere something rolls around a closed curve while its own center travels a loop rather than a straight line, coins included but far from unique to them.