Coin Rotation Paradox Calculator
How many times does a coin rotate when rolling all the way around another coin? Intuition says R/r (path length ÷ own circumference), but the real answer is R/r + 1. Enter the two radii to see the paradox in numbers.
True number of times the rolling coin rotates on its own axis
- 1
Naive rotations (path ÷ circumference)
R ÷ r = 3 ÷ 1 = 3 - 2
Extra rotation from revolution
1Orbiting the fixed coin adds exactly one rotation relative to a fixed observer. - 3
Actual rotations
3 + 1 = 4
How does this calculator work?
A coin of radius r rolling around a fixed coin of radius R makes R/r + 1 rotations — not the intuitive R/r. The extra rotation comes from the coin's own revolution around the fixed coin. Enter both radii above to see the naive vs. actual count and the circumference breakdown.
Formula
How this is calculated
When a coin of radius r rolls without slipping around the outside of a fixed coin of radius R, the rolling coin travels a path of length 2πR. Dividing by its own circumference 2πr gives R/r — the "path" rotations. But this only counts how much the coin spins relative to the line between the two centres. Because the coin also completes one full revolution around the fixed coin, it gains one additional rotation relative to a fixed external observer. The total is therefore R/r + 1.
This is the coin rotation paradox, sometimes called the gear-tooth paradox. It appears in gear-ratio problems: when a planet gear of radius r orbits a sun gear of radius R, its shaft rotates R/r + 1 times per orbit, not R/r. The same principle determines the Earth's sidereal day (23 h 56 m) versus the solar day (24 h): Earth makes one extra rotation per year relative to distant stars because it also orbits the Sun.
The formula assumes the rolling coin travels along the outside of the fixed coin with no slipping. For internal rolling (inside a larger circle, like a hypocycloid), the answer becomes R/r − 1.
Frequently asked questions
The extra rotation comes from the revolution itself. As the rolling coin orbits the fixed coin it rotates once in the same direction relative to an outside observer, even if it were glued in place. Rolling contact adds R/r more rotations on top of that, giving R/r + 1 total.
Yes. In an epicyclic (planetary) gear set, the planet gear shaft rotates R/r + 1 times per orbit of the sun gear. The same geometry explains why Earth completes 366.25 sidereal days but only 365.25 solar days in one year.
For internal rolling (hypocycloid path), the formula becomes R/r − 1. When r = R/2, the inner coin traces a straight line (Tusi couple) and effectively makes 1 rotation per trip — though naively you would expect 2.
Also known as
TG we-Calculate Editorial Team. (2026). Coin Rotation Paradox Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/coin-rotation-paradox-calculator
TG we-Calculate Editorial Team. "Coin Rotation Paradox Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/coin-rotation-paradox-calculator.
TG we-Calculate Editorial Team, "Coin Rotation Paradox Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/coin-rotation-paradox-calculator
@misc{wecalculate_coin_rotation_paradox_calculator, title = {Coin Rotation Paradox Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/coin-rotation-paradox-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?