Coin Rotation Paradox Calculator
When one coin rolls around another without slipping, it completes more rotations than you expect. Enter the radius of the fixed (stationary) coin and the rolling coin below. The calculator tells you exactly how many full rotations occur, shows you the step-by-step derivation, and explains why the result is always one more (or one fewer) than simple circumference division predicts.
Formula
Worked example
A US quarter (radius 12.13 mm) rolls around a US half-dollar (radius 15.09 mm): ratio = 15.09 / 12.13 = 1.244, total rotations = 1.244 + 1 = 2.244. For two identical quarters (radius 12.13 mm each): ratio = 1, total = 1 + 1 = 2. This is the classic paradox: even though the circumferences are equal and the rolling coin covers a distance equal to its own circumference, it rotates twice, not once.
What is the coin rotation paradox?
The coin rotation paradox is the counterintuitive result you get when one coin rolls without slipping around the outside of another coin of equal size. Naive reasoning says the rolling coin should rotate exactly once, because it travels a distance equal to its own circumference. The actual answer is two rotations. The extra rotation comes not from rolling along the perimeter but from the circular shape of the path itself. As the rolling coin makes one complete orbit around the fixed coin, that orbital motion contributes one full rotation relative to a fixed external reference frame. The paradox earned lasting fame in 1982 when it appeared on the SAT and the official answer key turned out to be wrong; three students challenged it and the test was re-graded.
How to use this calculator
Enter the radii of the fixed coin and the rolling coin in any unit you prefer (the result is unit-independent). Choose external rolling (the rolling coin goes around the outside, which is the classic setup) or internal rolling (the rolling coin travels inside a ring or annulus, used in planetary gear systems). The calculator shows the total rotation count, the naive expectation from simple circumference division, the +1 or -1 orbital contribution that creates the paradox, and a chart of how the rotation count varies across different radius ratios. The "show your work" panel walks through the derivation step by step with your actual numbers.
Why does the paradox happen? The geometry explained
Imagine "unrolling" the fixed coin so its curved rim becomes a straight line. The rolling coin travels along this line and rotates exactly once relative to the line, matching naive intuition. Now re-roll it back into a circle. The rolling coin must also orbit around the center of the fixed coin as it travels - and completing one full orbit around any point adds one full rotation relative to a stationary observer outside. This is sometimes called the "revolution paradox" and it applies to any closed path, not just circles. A coin rolling around a square still picks up exactly one extra rotation from the corners, and a planet orbiting a star makes one more rotation relative to distant stars than it does relative to the star itself - that is why Earth has a sidereal day slightly shorter than its solar day.
The general formula and internal rolling
For external rolling, the center of the rolling coin traces a circle of radius R + r, where R is the fixed radius and r is the rolling radius. Its circumference is 2*pi*(R + r). Dividing by the rolling coin circumference 2*pi*r gives (R + r)/r = R/r + 1 total rotations. For internal rolling (the rolling coin moves inside a ring), the center path radius is R - r and the formula becomes R/r - 1. Internal rolling is the geometry inside epicyclic gear trains and planters: a planet gear rolling inside a ring gear with R = 2r completes exactly 1 rotation per orbit rather than the 2 rotations that external rolling would give. When r approaches R in internal mode, the rotation count approaches zero, and when r = R exactly, internal rolling is impossible because the rolling coin would fill the entire ring.
Classic radius ratio examples
| R / r ratio | Naive expectation (R/r) | Actual rotations (R/r + 1) | Example |
|---|---|---|---|
| 1 | 1 | 2 | Two identical coins (classic paradox) |
| 2 | 2 | 3 | Fixed coin twice the rolling coin size |
| 3 | 3 | 4 | Fixed coin three times the rolling coin size |
| 4 | 4 | 5 | Fixed coin four times the rolling coin size |
| 1/2 | 0.5 | 1.5 | Rolling coin twice the size of fixed coin |
| 1/3 | 0.333... | 1.333... | Rolling coin three times the size of fixed coin |
Total rotations for common radius ratios in external rolling mode. The actual count always exceeds the naive circumference ratio by exactly 1.
Frequently asked questions
Why does a coin rolling around an identical coin rotate twice?
When the two coins have the same radius, the rolling coin travels a distance equal to the fixed coin circumference, which would produce one rotation if the path were straight. But the path is curved: it forms a circle around the fixed coin. Completing one full orbit around any point adds exactly one extra rotation relative to a fixed external observer. One rotation from rolling plus one from orbiting equals two total rotations.
What is the formula for the number of rotations?
For a coin of radius r rolling around the outside of a fixed coin of radius R, the total rotations are N = R/r + 1. For internal rolling (inside a ring of radius R), the total is N = R/r - 1. The "+1" and "-1" terms represent the orbital contribution from the curved path; without this correction you only get the rolling contribution R/r, which is always off by exactly 1 from the true result.
Does the paradox apply to coins of different sizes?
Yes. The formula R/r + 1 works for any pair of radii. For example, a coin with radius r rolling around a coin with radius 3r rotates 3/1 + 1 = 4 times, not the naive 3. The paradox term (+1) never changes regardless of the size ratio.
What happens in internal (annular) rolling?
When the rolling coin travels inside a ring, the center path radius shrinks to R - r rather than growing to R + r. The formula becomes N = R/r - 1, so the orbital contribution is -1 instead of +1. This describes the motion of planet gears in epicyclic (planetary) gear trains. For two equal coins in internal mode, rolling is geometrically impossible (the coin fills the ring). For R = 2r, N = 1: the planet rotates exactly once per orbit.
Is the coin rotation paradox related to Earths sidereal day?
Yes, exactly. Earth's sidereal day (23 hours 56 minutes) is the time for one rotation relative to distant stars, while the solar day (24 hours) is relative to the Sun. Because Earth also orbits the Sun once per year, it makes one extra rotation per year relative to distant stars - 366 sidereal days per year compared to 365 solar days. This is mathematically identical to the coin rolling around a fixed coin: the orbital contribution always adds one rotation over a complete orbit.
What did the 1982 SAT mistake have to do with this paradox?
In May 1982 the SAT included a problem asking how many times coin A rotates while rolling around coin B of the same size. The official answer was 1. Three high-school students independently identified that the correct answer is 2. The College Board re-graded the test, affecting roughly 300,000 students. The mistake became a famous example of the paradox's power to mislead even careful question-writers.