Coin Rotation Paradox Calculator

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Choose the unit for entering coin radii. The rotation count is unit-independent.
The radius of the stationary coin (half its diameter). A US quarter is about 12.13 mm radius.
mm
The radius of the rolling coin. Set equal to the fixed coin to see the classic paradox result of 2.
mm
External: the rolling coin travels around the outside. Internal: it rolls inside a larger ring. Internal mode requires the fixed coin to be larger than the rolling coin.
Total rotationsWhole number result
3

Number of times the rolling coin spins relative to a fixed external observer

Rolling rotations only2
Orbital contribution1
Center path circumference228.65mm
Radius ratio (R / r)2
Naive expectation (R / r)2
Paradox difference1
Naive expectation (R / r)2
Actual rotations (the paradox result)3

The rolling coin completes 3.0000 full rotations - not 2.0000 as naive reasoning suggests.

  • The rolling coin makes 3.0000 total rotations as it completes one full loop around the outside of the fixed coin.
  • Simple circumference division (R / r = 2.0000) predicts 2.0000 rotations, but the actual total is always one more rotation than that.
  • The extra +1 term comes from the curvature of the path: as the rolling coin orbits once around the fixed coin, that orbit itself contributes exactly one complete rotation relative to a fixed observer.
  • The center of the rolling coin traces a circle of circumference 228.65 mm. Dividing by the rolling coin circumference (76.22 mm) gives exactly 3.0000 rotations.

Next stepTry setting both radii equal to see the classic paradox: a coin rolling around an identical coin rotates twice, not once.

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