About

A coin of radius r rolling without slipping around a fixed coin of radius R completes Rr + 1 full self-rotations per orbit. Most people predict Rr rotations by analogy with rolling on a flat surface. The extra rotation arises because the rolling coin's center traverses a circular path of circumference 2π(R + r), not 2πR. This is identical to the distinction between sidereal and synodic periods in orbital mechanics. The paradox appeared on the 1982 SAT exam and was answered incorrectly by the test authors themselves.

This simulator renders the geometry in real time. The marker dot on the rolling coin tracks cumulative self-rotation while the traced curve reveals the epicycloid. Adjust the radius ratio to observe that equal-sized coins yield 2 rotations, not 1. For interior rolling (a coin rolling inside a larger coin), the count becomes Rr − 1. The tool assumes ideal no-slip contact and zero-thickness edges.

Formulas

The rolling coin's center sits at distance R + r from the origin. At revolution angle θ (measured in radians around the fixed coin), the center coordinates are:

x = (R + r) cos θ

y = (R + r) sin θ

The no-slip constraint means the arc lengths on both coins must match. The arc on the fixed coin is Rθ. Dividing by the rolling coin's radius gives the rolling-only rotation angle:

φ_roll = Rr θ

But the coin also revolves. An observer in the lab frame sees the coin rotate by its rolling angle plus the revolution angle. The total self-rotation in the lab frame:

φ_total = R + rr θ

After one full orbit (θ = 2π), the number of full rotations N is:

N = φ_total2π = R + rr = Rr + 1

A point on the rim of the rolling coin traces an epicycloid. The parametric equations for the traced path are:

x_trace = (R + r) cos θ − r cos(R + rr θ)

y_trace = (R + r) sin θ − r sin(R + rr θ)

Where R = radius of the fixed coin, r = radius of the rolling coin, θ = revolution angle (radians), φ_total = total self-rotation angle, N = number of complete self-rotations per orbit.

Reference Data

Radius Ratio R : r	Intuitive Guess (Flat)	Actual Rotations (External)	Actual Rotations (Internal)	Epicycloid Type	Cusps
1 : 1	1	2	0 (slides)	Cardioid	1
2 : 1	2	3	1	Nephroid	2
3 : 1	3	4	2	3-cusped epicycloid	3
4 : 1	4	5	3	4-cusped epicycloid	4
5 : 1	5	6	4	5-cusped epicycloid	5
1 : 2	0.5	1.5	-	Ellipse-like loop	-
2 : 3	0.667	1.667	-	Non-integer loop	-
3 : 2	1.5	2.5	0.5	Epitrochoid	-
7 : 3	2.333	3.333	1.333	Non-integer loop	-
10 : 1	10	11	9	10-cusped epicycloid	10
1 : 1 (flat line)	1	1 (no orbit bonus)	-	Straight line	0
R : r (general)	Rr	Rr + 1	Rr − 1	Epicycloid / Hypocycloid	Rr

Frequently Asked Questions

On a flat surface, a coin of circumference 2πr rolling along a distance 2πR rotates exactly R/r times. But on a curved surface, the coin's center travels along a circle of radius R + r, giving a path length of 2π(R + r). Dividing by the coin's circumference yields (R + r)/r = R/r + 1. The additional rotation is a topological consequence of orbiting a closed loop: even a wheel rolling around a point (zero-radius circle) would complete exactly 1 rotation.

Yes. For internal rolling (hypocycloid motion), the center travels a circle of radius R − r. The rotation count becomes R/r − 1. In the special case where R = 2r, the inner coin rotates exactly once and traces a straight line (diameter) - this is the basis of the Tusi couple mechanism used in medieval astronomy.

The Earth completes approximately 366.25 sidereal rotations per year but only 365.25 solar days. The missing day is consumed by the orbital revolution. This is the same +1 phenomenon. From the Sun's perspective (inertial frame), the Earth rotates once more than we experience from the surface. The coin rolling paradox is a tabletop version of this astronomical fact.

When R/r is irrational, the rolling coin never returns to its exact starting configuration. The traced epicycloid never closes and eventually fills an annular region densely (an ergodic path). When R/r = p/q (a rational number in lowest terms), the path closes after q full orbits with p cusps. The simulator traces multiple orbits to demonstrate this.

In the 1982 SAT, question 17 asked how many rotations a circle of radius 1 makes rolling around a circle of radius 3. The intended answer was 3, but the correct answer is 4. After challenges from students and teachers, the College Board rescinded the question and adjusted scores for approximately 300,000 test-takers. It remains one of the most famous standardized-test errors.

This model assumes zero-thickness disks and perfect no-slip rolling. Real coins have thickness that lifts the contact point, and friction is finite. For physical coins, micro-slipping at the contact point reduces the observed rotation count slightly below the theoretical value. The deviation is typically under 2% for smooth, rigid coins on a flat table and increases with softer materials.