Coin Rotation Paradox Calculator (how many turns a rolling coin makes)

Understanding the coin rotation paradox calculator — concept and principle

This calculator answers the coin rotation paradox: when one coin rolls without slipping once around another, how many times does it turn? For two equal coins rolling around the outside, the surprising answer is 2, not 1. Enter any two radii and choose inside or outside to see the general rule.

The rolling coin turns once for the distance and once more for circling the centre.

How it is calculated

Rolling around the outside: N = (R + r) ÷ r = R/r + 1

Rolling around the inside: N = (R − r) ÷ r = R/r − 1

Path of the rolling coin's centre = 2π(R + r) outside, or 2π(R − r) inside

Length of the contact path = 2πR

R is the radius of the fixed coin and r the radius of the rolling coin, in the same unit.

The principle

Rolling without slipping means the rolling coin's rim unrolls along the fixed coin's rim, which alone would turn it R ÷ r times. But the rolling coin is also carried once around the fixed coin, like the Moon around the Earth, and that adds one full turn when it goes around the outside, or removes one when it goes around the inside. An equivalent way to see it: the centre of the rolling coin travels a circle of radius R + r, and the coin turns once for every 2πr its centre travels.

Worked example

Default inputs: R = 12 mm, r = 12 mm, around the outside.

Turns from the rolling distance alone = R ÷ r = 12 ÷ 12 = 1.
Going around adds +1 turn.
Total turns = (12 + 12) ÷ 12 = 2.
Path of the centre = 2π × 24 = 150.796 mm; contact path = 2π × 12 = 75.3982 mm.

Each point is the calculator re-run with only “Radius of the rolling coin r (mm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Radius of the rolling coin r (mm)	Turns of the rolling coin
3	5
6	3
9	2.33333
12	2
15	1.8
18	1.66667
24	1.5

How to read the result

The "+1" or "−1" is the effect of the trip around the fixed coin, independent of the sizes. For a small coin rolling around a big one, the extra turn matters little; for equal coins it doubles the count. The same reasoning explains why a sidereal day is shorter than a solar day: Earth's trip around the Sun changes the count of turns by one per year.

Common mistakes

Counting only the contact distance and forgetting the trip around.
Adding the turn for inside rolling instead of subtracting it.
Mixing radius and diameter.
Allowing the coin to slip, which breaks the rule.

Keep in mind: The result assumes perfect rolling without slipping and perfectly round coins.

Frequently asked questions

How many times does a coin rotate around an identical coin?

Twice when it rolls around the outside.

What if it rolls inside a ring twice its size?

N = (2r − r) ÷ r = 1 turn.

Where else does this effect appear?

In gear trains with planetary gears and in the difference between sidereal and solar days.

Sources: O. L. Santos-Pereira (2025), "Roullete curves, coin paradox and Aristotle's wheel paradox", arXiv:2512.00123

Checked 9 Nov 2026. O. L. Santos-Pereira, “Roullete curves, coin paradox and Aristotle’s wheel paradox”, arXiv:2512.00123 (2025) — a circle of radius r rolling without slipping around a fixed circle of radius R turns N = (R ± r)/r times (+ outside, − inside); equal coins give 2 turns.

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