Coin Rotation Paradox

The formulas

rolling turns = R/r + 1 (outside), R/r − 1 (inside)

The paradox

Roll a coin once around another coin of the same size and it turns twice, not once — even though the path is exactly one circumference long. Dividing path length by circumference gives 1, and that answer is wrong.

The missing turn comes from the frame of reference: the coin rotates once relative to the line joining the centres, and that line itself rotates once as it goes around. Rolling around the inside subtracts a turn instead, for the same reason with the sign flipped.

Everything follows from the radius

A circle has one degree of freedom, so any single measurement fixes all the others:

• d = 2r
• C = 2πr = πd
• A = πr² = πd²/4 = C²/4π

Going backwards, r = C/2π and r = √(A/π). The calculator above accepts whichever you happen to have.

Area grows with the square

Double the radius and the circumference doubles, but the area quadruples. That is why a 16-inch pizza is more than twice the food of a an 11-inch one, and why pipe capacity rises so sharply with bore.