Explore the coin rotation paradox: how many times does a coin rotate when rolling around another? Handles inner/outer rolling, different radii, path length, and demonstrates the paradox with visuals.
The coin rotation paradox is a delightful mathematical surprise. When a coin of radius r rolls around the outside of a fixed coin of radius R without slipping, most people guess it makes R/r rotations. But the actual answer is R/r + 1 for outer rolling, because the rolling coin also revolves once around the center of the fixed coin. For inner rolling (inside a fixed ring), it is R/r − 1 (or |R/r − 1|) rotations. This extra (or missing) rotation is analogous to the difference between sidereal and solar days in astronomy — the Earth rotates 366.25 times per year relative to the stars, not 365.25, because of its revolution around the Sun. This calculator lets you set any radii for both coins, toggle between inner and outer rolling, and see the total rotations, the path length traveled by the center, the circumference ratio, and the "paradox correction" that accounts for the revolution. A rolling animation visual shows the coin tracing its path, and a demonstration table computes rotations for common radius ratios. Preset examples include the classic equal-coin problem (answer: 2, not 1), as well as cases with 2:1, 3:1, and arbitrary ratios.
The coin rotation paradox is notoriously counterintuitive — most people get it wrong even after hearing the explanation. This calculator makes the paradox tangible by showing the contact rotations and revolution rotation separately, comparing the naive guess (R/r) with the actual answer (R/r ± 1), and visualizing the rolling path. It handles both inner and outer rolling, partial revolutions, and arbitrary radius ratios, making it ideal for math enrichment, competition training, and physics demonstrations of sidereal vs. solar rotation.
Outer rolling: Rotations = R/r + 1, Path = 2π(R + r) Inner rolling: Rotations = |R/r − 1|, Path = 2π|R − r| Circumference ratio = R/r
Result: 2 rotations
Two equal coins (R = r = 1): Outer rolling → 1/1 + 1 = 2 full rotations. The small coin rotates twice, not once! Path = 2π(1+1) = 4π.
When a coin of radius r rolls around one of radius R, the naive guess is R/r rotations — the ratio of circumferences. But this only counts rotations due to surface contact. The rolling coin also revolves once around the center of the fixed coin, adding (for outer rolling) or subtracting (for inner rolling) one full rotation. The total is R/r + 1 for outer rolling and |R/r − 1| for inner rolling. For equal coins (R = r), the naive guess is 1 but the actual answer is 2 — the coin rotates twice!
The coin rotation paradox is mathematically identical to the difference between sidereal and solar days. Earth rotates 366.25 times per year relative to the stars (sidereal days) but only 365.25 times relative to the Sun (solar days). The "missing" day comes from Earth's revolution around the Sun — exactly the same extra rotation a coin picks up when circling another coin. This connection makes the paradox a powerful teaching tool for astronomy concepts.
When a small circle rolls outside a larger one, the path traced by a point on the rolling circle is called an epicycloid. When rolling inside, it is a hypocycloid. Special cases include the cardioid (R = r, outer), the circle (R = 2r, inner, which traces a straight diameter — the Tusi couple), and the astroid (R = 4r, inner). These curves appear in gear design (epicyclic gearing), Spirograph toys, and even in Ptolemy's ancient astronomical models.
When a coin rolls around another same-sized coin, it makes 2 full rotations instead of the expected 1. The extra rotation comes from its revolution around the center.
The coin both spins on its own axis (R/r turns from contact rolling) AND revolves once around the fixed coin. These combine to R/r + 1 total rotations.
For inner rolling, the revolution subtracts instead of adding: total rotations = R/r − 1 (for R > r). Use this as a practical reminder before finalizing the result.
Yes! Earth makes ~366.25 sidereal rotations per year but ~365.25 solar days. The extra rotation from orbiting the Sun is the same principle.
Equal radii is the classic case: outer rolling gives 2 rotations, inner rolling gives 0 rotations (the coin translates without spinning). Keep this note short and outcome-focused for reuse.
The paradox applies to any convex closed path, not just circles. For a circle of circumference C, a coin of circumference c makes C/c + 1 rotations (outer).