Coin Rotation Paradox Calculator

What Is the Coin Rotation Paradox?

The Coin Rotation Paradox is a famous counter-intuitive problem in geometry and kinematics. When one coin rolls without slipping around the perimeter of an identical stationary coin, intuition suggests it should complete exactly 1 rotation around its own center, since both coins have identical circumferences ($2\pi r$). However, in reality, the rolling coin completes 2 full rotations ($720^\circ$).

This phenomenon gained widespread fame in 1982 when it appeared on the SAT exam. The College Board presented a multiple-choice question asking how many rotations a coin with radius $r$ makes while rolling around a stationary coin of radius $3r$. They listed $\frac{3r}{r} = 3$ as the correct answer. In truth, the rolling coin performs $\frac{3r}{r} + 1 = 4$ rotations. A high school student noticed the error, forcing the testing service to rescore over 300,000 exams nationwide.

The Mathematical Formula for Coin Rotation

To understand why the extra rotation occurs, consider both the intrinsic rotation (rolling distance along the perimeter) and the extrinsic rotation (the circular path of the coin center orbiting the stationary center):

1. Rolling Around the Outside (Epicycloid Motion)

When a coin of radius $r$ rolls around the outside of a stationary circle of radius $R$:

$$N = \frac{R}{r} + 1$$

Here:

$\frac{R}{r}$: The number of rotations resulting from the perimeter rolled (intrinsic turning).
$+1$: The additional full rotation resulting from tracing a $360^\circ$ circular orbit around the central coin (extrinsic turning).

2. Rolling Around the Inside of a Hollow Ring (Hypocycloid Motion)

When a smaller coin of radius $r$ rolls along the inside of a hollow cylinder or ring of radius $R$ ($R > r$):

$$N = \frac{R}{r} - 1$$

Because the inner orbit turns in the reverse direction relative to the contact surface, the orbit subtracts $1$ rotation instead of adding one.

3. Rolling Along a Straight Line

When a coin rolls along a straight flat track of length $L$, there is no curved orbit ($N_{\text{extrinsic}} = 0$). Thus, the number of rotations is simply:

$$N = \frac{L}{2\pi r}$$

Detailed Comparison Table

Scenario	Condition	Formula ($N$)	Example ($R=10, r=10$)
Outer Roll (Epicycloid)	Rolling outside circle	$$N = R/r + 1$$	$$10/10 + 1 = \mathbf{2 \text{ rotations}}$$
Inner Roll (Hypocycloid)	Rolling inside ring ($R > r$)	$$N = R/r - 1$$	$$30/10 - 1 = \mathbf{2 \text{ rotations}}$$
Straight Line	Rolling on flat line ($L = 2\pi R$)	$$N = L / (2\pi r)$$	$$20\pi / 20\pi = \mathbf{1 \text{ rotation}}$$

Why Does the Extra Rotation Happen?

The illusion arises because people imagine the coin rolling along an unrolled straight string of length $2\pi R$. While the contact point travels a distance of $2\pi R$ along the surface, the center of the moving coin travels along a larger circle with radius $R_{\text{path}} = R + r$.

The total distance traveled by the center of the rolling coin is:

$$D_{\text{center}} = 2\pi (R + r) = 2\pi R + 2\pi r$$

Dividing this total center displacement by the coin's circumference ($2\pi r$) yields:

$$\frac{2\pi (R + r)}{2\pi r} = \frac{R + r}{r} = \frac{R}{r} + 1$$

For more geometric tools, explore our Circle Theorems Calculator and Chord Length Calculator.

Frequently Asked Questions

Why is the coin rotation paradox considered a paradox?

It is a veridical paradox: a result that is true and mathematically provable, but completely defies everyday human intuition. Most people mistakenly assume that because the circumferences of identical coins are equal, one coin must complete exactly one turn.

What was the 1982 SAT math question error?

In May 1982, an SAT math question asked for the number of rotations a coin of radius $r$ makes around a stationary circle of radius $3r$. The official choices did not even include the correct answer (4 rotations), because the test makers incorrectly calculated $\frac{3r}{r} = 3$.

Does the coin rotation paradox apply to gears and planetary gearboxes?

Yes. The exact same kinematic principles govern epicyclic (planetary) gear systems used in automotive automatic transmissions, wind turbines, and robotics. Engineers must account for the $+1$ orbital rotation when calculating gear ratios.

What happens if the moving coin is 3 times larger than the stationary circle?

If $R = 1$ and $r = 3$, the formula $N = \frac{R}{r} + 1$ yields $N = \frac{1}{3} + 1 = \frac{4}{3} = 1.3333$ rotations.

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