How this calculator works
Roll a coin around the outside of an identical coin and it turns twice before returning to its starting orientation. The naive answer is once, because the two circumferences are equal. The missing turn comes from the path itself: the rolling coin goes once around the fixed coin, so its own frame rotates once as it travels, in addition to the rotation produced by rolling. For a fixed radius R and a rolling radius r, rolling contributes R/r turns and going around the fixed coin contributes one more, giving R/r + 1. If the coin rolls around the inside of a ring, the travelling turn opposes the rolling turn, so the count is R/r - 1. When the coins are equal on the inside path, the two effects cancel and the coin keeps the same face toward the centre, making zero turns.
Formula
Outside path: N = (R + r)/r = R/r + 1. Inside path: N = (R - r)/r = R/r - 1. In terms of diameters, R/r = D_fixed/D_rolling, so outside N = D_fixed/D_rolling + 1 and inside N = D_fixed/D_rolling - 1.
Worked example
Two equal coins of diameter 2 cm: the outside path gives 2/2 + 1 = 2 rotations, which is the classic surprise. A fixed coin of diameter 6 cm with a rolling coin of 2 cm gives 6/2 + 1 = 4 rotations outside. On the inside of a ring of diameter 4 cm, a 2 cm coin makes 4/2 - 1 = 1 rotation, and two equal coins make zero.
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Frequently asked questions
Why do two equal coins produce two rotations?
One rotation comes from rolling a distance equal to the fixed coin circumference. The other comes from carrying the rolling coin once around the fixed coin. Both add because the coin is on the outside, where travelling and rolling turn the same way.
Why is the inside answer one less?
On the inside of a ring the rolling direction opposes the direction of travel around the centre. The travelled turn cancels one rolled turn, so equal coins produce zero rotations and the same face stays toward the centre.
Is this the same as the coin rolling on a straight line?
No. On a straight line a coin of radius r makes one turn per 2πr of travel. The extra term here comes from the path curving around the fixed coin, not from the rolling itself.
What if the rolling coin is larger than the fixed coin?
The outside formula still works. On the inside path a larger rolling coin cannot fit around a smaller fixed coin without overlapping, so the calculator rejects that case.