Coin Rotation Paradox Calculator

Calculate how many times a rolling coin rotates as it moves around another coin of any radius — solving the famous coin rotation paradox.

Formula

Absolute rotations = (r₁ + r₂) ÷ r₂; Relative = r₁ ÷ r₂

The paradox: if both coins are the same size, the rolling coin makes 2 full rotations (not 1).
Relative rotations = circumference traveled / rolling coin circumference = r1/r2.
Absolute (in space) rotations = relative + 1 (one extra rotation from the orbit itself).

Inputs

Relative = 25/25 = 1. Absolute = (25+25)/25 = 2. The rolling coin makes 2 full rotations!

Why does the rolling coin rotate more than expected?

When a coin rolls around another, it gains an extra rotation from its orbital revolution around the center coin.

Is this related to the SAT problem?

Yes — this was famously a 1982 SAT problem whose official answer was wrong. The paradox was rediscovered.

Does this apply to gears?

Yes — planetary gear systems must account for this extra rotation in their gear ratios.

What if the coin rolls inside another circle?

For internal rolling (coin inside a larger circle), rotations = (r1 - r2) / r2 in the absolute frame.