Coin Rotation Paradox Calculator
Calculate coin rotation paradox instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Coin Rotation Paradox Calculator
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Formula: Rotations = R/r + 1 (outside) or R/r - 1 (inside)
Worked example — 4 rotations (not 3 as naively expected - this was the 1982 SAT error)
Formula
Rotations = R/r + 1 (outside) or R/r - 1 (inside)
When a coin of radius r rolls around a stationary coin of radius R, the total rotations equal the circumference ratio R/r plus 1 (for outside rolling) or minus 1 (for inside rolling). The extra rotation comes from the orbital revolution around the stationary coin.
Worked Examples
Example 1: Classic SAT Problem - Radius 3 and 1
Problem:A coin of radius 1 rolls around the outside of a coin of radius 3 without slipping. How many times does it rotate?
Solution:Stationary coin radius R = 3, rolling coin radius r = 1 Naive answer (circumference ratio): R/r = 3/1 = 3 rotations Actual answer (with revolution): R/r + 1 = 3 + 1 = 4 rotations The rolling coin travels a path of circumference 2pi(R+r) = 2pi(4) = 8pi Its own circumference is 2pi(1) = 2pi Center path / own circumference = 8pi / 2pi = 4 rotations
Result:4 rotations (not 3 as naively expected - this was the 1982 SAT error)
Example 2: Equal Coins Rolling Inside
Problem:A coin of radius 2 rolls inside a coin of radius 2. How many rotations does it make?
Solution:R = 2, r = 2, rolling inside Rotations = R/r - 1 = 2/2 - 1 = 0 rotations The rolling coin translates in a circle without rotating! Center path circumference = 2pi|R-r| = 2pi(0) = 0 The center of the rolling coin stays at the center of the stationary coin.
Result:0 rotations - the coin translates without rotating (special case)
Frequently Asked Questions
What is the coin rotation paradox?
The coin rotation paradox is a counterintuitive phenomenon that occurs when one coin rolls around another without slipping. When a coin with radius r rolls completely around the outside of a stationary coin with radius R, it makes R/r + 1 rotations about its own center, not R/r as most people expect. The extra rotation comes from the fact that the rolling coin also orbits around the stationary coin once, contributing one additional self-rotation. For example, when two identical coins are used (R = r), the rolling coin makes 2 full rotations, not 1. This paradox gained widespread attention in 1982 when a question about it appeared on the SAT, and all the provided answer choices were wrong because the test makers had overlooked the extra rotation.
Why does the rolling coin make an extra rotation?
The extra rotation occurs because there are two separate contributions to the rolling coin's rotation. First, there is the rotation due to the rolling contact between the two surfaces: the coin rolls along an arc equal to the stationary coin's circumference, producing R/r rotations from friction alone. Second, there is the rotation due to revolution: as the rolling coin orbits around the stationary coin, it makes one complete loop, which contributes exactly one additional rotation relative to an external observer. Think of it this way: if you carried a non-rotating coin around a circle and brought it back to the start, it would have rotated once relative to the ground even without any rolling. The total rotation is the sum of both contributions.
How does rolling inside differ from rolling outside?
When a coin rolls inside a larger stationary coin (internal rolling or hypocycloid motion), the revolution effect subtracts rather than adds one rotation. The total rotations become R/r - 1 instead of R/r + 1. This is because the rolling coin orbits in the same direction as it rotates from contact, effectively canceling one revolution. A remarkable special case occurs when R = r: rolling inside gives zero rotations, meaning the inner coin translates in a circle without ever rotating about its own center. Another special case is R = 2r: the inner coin makes exactly one rotation while tracing a straight-line diameter (this is the basis of the Cardan gear mechanism). These internal rolling patterns produce beautiful curves called hypocycloids.
What is the connection to the 1982 SAT question?
In 1982, the SAT included a multiple-choice question asking how many times a smaller circle (radius 1) would revolve around a larger circle (radius 3) when rolling along the outside. The intended answer was 3 (the ratio of circumferences), and this was listed among the choices. However, the correct answer is 4, because of the extra rotation from revolution. None of the five answer choices was correct, which was pointed out by a high school student who wrote to the College Board. After verification, the College Board had to rescore the test for over 300,000 students. This incident became one of the most famous errors in standardized testing history and brought widespread attention to the coin rotation paradox.
How does this relate to sidereal versus synodic periods in astronomy?
The coin rotation paradox has a direct astronomical analogue. Earth makes about 365.25 solar days (synodic rotations relative to the Sun) in one year, but it actually rotates 366.25 times relative to the fixed stars (sidereal rotations). The extra sidereal day comes from Earth orbiting the Sun once, exactly analogous to the extra rotation of the rolling coin. If Earth did not rotate at all relative to the stars, it would still appear to rotate once per year as seen from the Sun (synodic effect). Similarly, the Moon always shows the same face to Earth (1:1 spin-orbit resonance), meaning it rotates exactly once per orbit relative to the stars. These astronomical phenomena are governed by the same mathematics as the coin rotation paradox.
What mathematical curves are traced by the coin rotation paradox?
When a point on the rolling coin is tracked as it rolls around the stationary coin, it traces a curve called an epicycloid (for external rolling) or a hypocycloid (for internal rolling). The specific curve depends on the radius ratio. For external rolling with R/r = 1 (equal coins), the traced curve is a cardioid. For R/r = 2, it is a nephroid. For R/r = 3, it is a three-cusped epicycloid. For internal rolling, R/r = 3 produces a deltoid (three-pointed star), and R/r = 4 produces an astroid (four-pointed star). These curves have important applications in gear design, where epicycloidal gear tooth profiles provide smooth, constant-velocity power transmission. Spirograph toys create these curves mechanically.
How is the coin rotation paradox used in mechanical engineering?
The principles behind the coin rotation paradox are fundamental to planetary gear systems (epicyclic gears) used in automatic transmissions, bicycle hub gears, and wind turbine gearboxes. In a planetary gear set, planet gears roll around a central sun gear, and the rotation count follows the same math as the coin paradox. The gear ratio depends on whether the ring gear, sun gear, or planet carrier is held fixed. Wankel rotary engines also rely on this principle: the triangular rotor makes epicycloidal motion inside the housing, with the rotor spinning at one-third the speed of the eccentric shaft due to the internal rolling geometry. Understanding the extra rotation is essential for correctly calculating gear ratios in these systems.
Can this paradox be generalized to non-circular shapes?
Yes, the coin rotation paradox generalizes to any convex shape rolling around another. The key insight is that the total rotation equals the rotation from contact (related to the arc length ratio) plus the rotation from revolution (related to the total turning of the path). For a coin rolling along a straight line, there is no revolution, so the rotations equal the distance divided by the circumference, as expected. For a coin rolling around any closed convex curve, it gains one extra rotation per complete trip, regardless of the curve's shape. For a coin rolling around a polygon, the extra rotation comes from the turns at the vertices. A coin rolling around a triangle (total exterior angle 360 degrees) still gains exactly one extra rotation, split among the three vertex turns.
What is the angular velocity relationship for rolling coins?
For a coin of radius r rolling without slipping around a stationary coin of radius R, the angular velocity relationship links the orbital angular velocity to the spin angular velocity. For external rolling, the spin angular velocity equals (R + r)/r times the orbital angular velocity. For internal rolling, it equals (R - r)/r times the orbital angular velocity. The total rotation rate relative to a fixed frame is the spin rate, which includes both the contact-driven rotation and the orbital contribution. In terms of linear velocity, the no-slip condition requires that the contact velocity be equal on both surfaces: r times the spin rate equals (R + r) times the orbital rate. These relationships are identical to those used in planetary gear train analysis.
How can you demonstrate the coin rotation paradox physically?
The easiest demonstration uses two identical coins (such as quarters). Place one coin flat on a table and roll the other coin around its edge without slipping, marking the starting orientation of the rolling coin. After one complete trip around the stationary coin, the rolling coin will have rotated twice, not once as intuition suggests. For a more controlled experiment, use cardboard circles with marked reference points and roll them carefully. You can also demonstrate the inside case by cutting a circle in a piece of cardboard and rolling a smaller disk inside it. Digital demonstrations can be created using geometry software like GeoGebra, which allows precise tracking of rotation angles. The physical demonstration is particularly compelling because seeing the two full rotations challenges most people's intuitive prediction of just one.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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