How the Coin Rotation Paradox works
When a coin rolls without slipping around the outside of another, fixed coin, it doesn't just spin according to the ratio of the two circumferences — it also revolves once around the fixed coin's center, and that revolution adds one extra full turn. This calculator finds the actual number of rotations for a coin of radius r rolling around a fixed coin of radius R, and compares it with the naive circumference-based guess.
Formula and method
Rolling around the outside of a fixed coin, one full lap gives:
- Actual rotations = (R + r) / r, where R is the fixed coin's radius and r is the rolling coin's radius.
- Naive estimate = R / r, based only on comparing the two circumferences.
- Difference = actual − naive = exactly 1, no matter what R and r are.
Rolling around the inside of a fixed ring (a hypocycloid path) flips the sign: actual rotations = (R − r) / r, exactly 1 fewer than the naive R / r estimate. This only works when the rolling coin's radius r is smaller than the ring's radius R. For multiple laps, multiply every rotation count by the number of laps.
Why the "extra" rotation happens
A rolling coin's total spin has two parts: the spin caused by rolling along the surface, and the spin caused by the rolling coin's center orbiting the fixed coin's center. Comparing circumferences only accounts for the surface-rolling part, so it misses the one full extra turn contributed by the orbit itself. This is the source of the coin rotation paradox — most famously, rolling one coin fully around an identical coin (R = r) produces 2 rotations, not the 1 you'd expect from equal circumferences.
Common sources of error
- Confusing R and r: R is the radius of the coin that stays still; r is the radius of the coin that rolls. Swapping them changes the ratio and the answer.
- Forgetting the extra rotation: the naive R / r ratio isn't the answer by itself — it's off by exactly 1 rotation per lap, in a direction that depends on whether the path is outside or inside.
- Inside path with r not smaller than R: rolling around the inside only makes sense when the rolling coin is smaller than the fixed coin or ring it travels inside.
Applications
The same (R ± r) / r relationship shows up in epicyclic (planetary) gear trains and in the combined orbit-and-spin motion of moons and satellites. It is also why a coin rolled once around a same-size coin ends up right-side up after two spins rather than one — a simple, physical demonstration of the difference between rotation measured against the ground and rotation measured against the rolling surface alone.