Coin Rotation Paradox Calculator

Understanding the Coin Rotation Paradox

The coin rotation paradox ranks among the most surprising results in elementary geometry. When a coin rolls around the outside of an identical coin without slipping, intuition suggests the rolling coin completes exactly one full rotation — after all, both coins share the same circumference. In reality, it completes two full rotations. This counter-intuitive result has fascinated mathematicians and students for decades, vividly demonstrating the critical difference between rotation measured relative to an external fixed frame and rotation measured relative to the path surface.

The Core Formula

The number of rotations N completed by a rolling coin is determined by:

N = (R ± r) / r

Where the variables are defined as:

• R — Radius of the fixed (stationary) circle, or the total path length in straight-line mode
• r — Radius of the rolling coin
• + (plus) — Applied when the coin rolls around the outside of the fixed circle, tracing an epicycloid path
• − (minus) — Applied when the coin rolls around the inside of the fixed circle, tracing a hypocycloid path

Three Rolling Modes Explained

1. External Rolling (Epicycloid)

When the rolling coin travels around the outside of the fixed circle, the center of the rolling coin traces a circle of radius R + r. The total arc length traversed by the contact point equals 2π(R + r). Since the rolling coin covers 2πr of arc per full rotation, the number of rotations is (R + r)/r. This is the mode that produces the famous paradox: setting R = r yields N = 2, not 1. As the Wikipedia article on the Coin Rotation Paradox documents, this result has a direct astronomical analogy — Earth's sidereal day differs from its solar day by exactly one rotation per year for the same geometric reason.

2. Internal Rolling (Hypocycloid)

When the rolling coin moves around the inside of a larger fixed circle, the effective travel radius shrinks to R − r, and the formula becomes N = (R − r)/r. For example, a coin of radius 5 mm rolling inside a fixed circle of radius 20 mm completes (20 − 5)/5 = 3 full rotations. According to MathWorld's Hypocycloid reference, special cases of internal rolling produce classic mathematical curves: when R = 4r the path is an astroid, and when R = 3r the path is a deltoid. These curves appear in mechanical linkages, lens-grinding machines, and architectural ornament design.

3. Straight-Line Rolling

When a coin rolls along a flat surface over a path of length L, the number of rotations equals L / (2πr). This baseline case produces no paradox — the result matches the simple ratio of path length to circumference. This mode provides a useful benchmark for understanding why the circular cases yield unexpected values: the extra rotation in circular rolling comes entirely from the coin's orbital revolution around the fixed center, an effect absent on a flat surface.

Why the Paradox Occurs

The paradox arises from conflating two distinct reference frames. In the local frame (measured relative to the path surface), the rolling coin rotates once for every full circumference of the fixed circle it traverses — a count of R/r. However, in the global fixed frame (measured by a stationary outside observer), the coin also completes one full revolution around the center of the fixed circle, adding exactly 1 to the total rotation count. The two contributions combine to give N = R/r + 1 = (R + r)/r for external rolling. For internal rolling the revolution runs in the opposite sense to the spin, yielding N = R/r − 1 = (R − r)/r instead.

Historical Context

The coin rotation paradox gained widespread public attention when a version appeared on the 1982 SAT (Scholastic Aptitude Test). The problem asked how many times a coin of radius r/3 would rotate while rolling around a larger coin of radius r. The naively expected answer was 3; the correct answer accounting for the revolution is 4. The College Board accepted both answers after the error was identified, illustrating how deeply counter-intuitive this result is even for professional test designers. The same geometric principle appeared in the historical debate between geocentric and heliocentric models of the solar system: the difference between a planet's sidereal and synodic periods reflects the identical extra-rotation effect at a planetary scale.

Worked Numerical Examples

Example 1: The Classic Paradox (Equal Coins)

Fixed coin radius R = 25 mm, rolling coin radius r = 25 mm, external mode: N = (25 + 25) / 25 = 2.0 rotations. The rolling coin makes twice as many rotations as naive intuition predicts.

Example 2: Small Roller on a Large Fixed Circle

Fixed circle radius R = 100 mm, rolling coin radius r = 10 mm, external mode: N = (100 + 10) / 10 = 11 rotations. The naive estimate of R/r = 10 is off by exactly one full rotation.

Example 3: Internal Rolling

Fixed circle radius R = 50 mm, rolling coin radius r = 10 mm, internal mode: N = (50 − 10) / 10 = 4 rotations.

Real-World Applications

• Astronomy: Earth's sidereal day (23 h 56 min 4 s) versus the solar day (24 h) represents the same extra-rotation effect at planetary scale — Earth gains one additional rotation per year relative to distant stars.
• Planetary gear systems: Engineers use the (R ± r)/r relationship to calculate output shaft rotation counts in epicyclic gearboxes found in automotive automatic transmissions and wind-turbine nacelles.
• Rotary offset printing: Cylinder-to-cylinder ink transfer systems must account for the paradox when computing pattern repeat lengths on matched impression and blanket cylinders.
• Spirograph mathematics: The classic Spirograph drawing toy generates hypocycloid and epicycloid curves using exactly this rolling-circle geometry, with different integer R/r ratios producing distinct petal counts in the resulting figures.