Calculate exact rotations of a rolling circle around a fixed circle (\(N = \frac{R}{r} + 1\) for outer epicyclic, \(N = \frac{R}{r} - 1\) for inner hypocyclic). Solve the 1982 SAT math error, sidereal vs solar day orbits, and planetary gear mechanics.
Select rolling configuration and specify circle dimensions.
1.000 Rolling Contact + 1.000 Orbital Rotation = 2.000 (720.0°)
| 🔄 Total Rotations (\(N\)) | 2.000 Turns (720.00°) |
| ⚙️ Perimeter Contact Rotations | 1.000 Turns (360.00°) |
| 🪐 Orbital Turn Contribution | +1.000 Turn (+360.00°) |
| ⭕ Stationary Circle Radius (\(R\)) | 10.000 mm |
| 🪙 Rolling Coin Radius (\(r\)) | 10.000 mm |
| 📏 Center Path Radius (\(R \pm r\)) | 20.000 mm (Path: 125.66 mm) |
| ⏱️ Angular Velocity Ratio | 2.000 × ω_orbit |
The Coin Rotation Paradox is one of the most deceptively simple yet universally misunderstood problems in geometry and classical kinematics. Humans intuitively assume that rolling contact distance alone determines rotation count. Our calculator solves critical analytical and educational challenges:
When people calculate rotations, they divide the stationary circumference by the rolling circumference (\(2\pi R / 2\pi r = R/r\)). This overlooks the fact that the rolling coin is simultaneously rotating as it traverses its curved circular orbit. Our tool breaks down the motion into exact contact turns and orbital turns so you can see why the extra \(+1\) rotation physically exists.
In automotive automatic transmissions, robotic actuators, and wind turbine gearboxes, planet gears revolve around central sun gears. Mechanical engineers use our tool to calculate precise kinematic speed ratios (\(\omega_{\text{planet}} = \omega_{\text{carrier}}(1 + R_{\text{sun}}/R_{\text{planet}})\)) without drawing complex velocity vector polygons.
Students and astrophysicists frequently struggle with why Earth completes 366.25 sidereal rotations during a 365.25-day solar year. Our calculator models this planetary orbital geometry directly, proving how orbital revolution adds an exact \(+1\) turn relative to the inertial celestial frame.
Since the famous 1982 SAT math question error, variations of rolling circle paradoxes frequently appear in AMC 10/12, Putnam, and physics olympiads. Our calculator provides instant visual verification for arbitrary radius ratios and partial-orbit sweeps.
Select Outside Circle (Epicyclic), Inside Ring (Hypocyclic), or Flat Straight Line to set your boundary geometry.
Input the stationary circle radius (\(R\)) and rolling circle radius (\(r\)), or click a preset like Identical Coins or the 1982 SAT Question.
Use the interactive slider to sweep through custom orbit angles (\(\phi\))—such as \(180^\circ\) for half-way or \(360^\circ\) for a full orbit.
Click Copy Coin Kinematics Card to copy a formatted calculation report to your clipboard for homework, reports, or engineering docs.
Full kinematic coverage for outer convex rolling (\(R/r + 1\)), inner concave ring rolling (\(R/r - 1\)), and straight flat line rolling (\(L / 2\pi r\)).
Sweep between \(0^\circ\) and \(720^\circ+\) to observe how orientation vectors rotate at intermediate orbital angles (e.g. \(1.0\) rotation at \(180^\circ\)).
All calculations execute client-side via JavaScript with zero server latency, zero tracking, and complete privacy.
The Coin Rotation Paradox is a counterintuitive mathematical truth in kinematics. When one coin rolls without slipping around another identical coin of the same radius (\(R = r\)), an external observer watching from above sees the rolling coin rotate 2 full turns (\(720^\circ\)) on its axis by the time it returns to its starting point—not 1 turn.
Rolling an identical coin around another coin of the same size: $$N = \frac{10 + 10}{10} = \frac{10}{10} + 1 = 1.0 + 1.0 = \mathbf{2.000\text{ full rotations (720}^\circ\text{)}}$$ Contact perimeter distance traversed is \(2\pi(10) \approx 62.83\,\text{mm}\), but the rolling center of mass travels a larger circle of radius \(10 + 10 = 20\,\text{mm}\) (\(C = 125.66\,\text{mm}\)).
A smaller circle of radius \(r\) rolling around a fixed circle of radius \(3r\): $$N = \frac{3r + r}{r} = \frac{4r}{r} = 3 + 1 = \mathbf{4.000\text{ full rotations (1440}^\circ\text{)}}$$ The rolling coin completes 3 rotations from surface rolling contact plus 1 rotation from its orbital loop around the center.
In the May 1982 SAT exam, question #17 asked: "A circle with radius \(r\) rolls without slipping around a fixed circle with radius \(3r\). How many revolutions does the smaller circle make when returning to its starting position?"
The College Board's Official Options: (A) \(3/2\) • (B) \(24/7\) • (C) \(3\) • (D) \(9/2\) • (E) \(6\)
The test writers naively computed \(\frac{2\pi(3r)}{2\pi r} = 3\) and declared (C) 3 as the correct answer. The actual answer was not even among the options!
Three test-takers—including Douglas Lusardi from Florida—spotted the error and proved that because the center of the rolling circle travels along a radius of \(3r + r = 4r\), the true circumference traveled is \(2\pi(4r)\), requiring exactly \(\frac{4r}{r} = 4\) full rotations. The College Board officially rescored 300,000 exams, adjusting thousands of students' SAT scores upward.
The Coin Rotation Paradox governs Earth's rotation around the Sun. To complete 1 solar day (noon to noon), Earth must rotate roughly \(360.986^\circ\) on its axis to compensate for its orbital movement along its path.
However, relative to the stationary distant stars (the cosmic "lab frame"), Earth rotates exactly \(360.000^\circ\) once every 23 hours, 56 minutes, and 4 seconds (1 sidereal day). Because Earth makes 1 full revolution around the Sun each year, it gains exactly +1 extra rotation, completing 366.25 sidereal rotations during 365.25 solar calendar days!
Rigorous kinematic decomposition comparing outer rolling vs. inner ring rolling.
| Configuration | Radius Ratio (\(R/r\)) | Contact Turns | Outer Rotations (\(N_{\text{out}}\)) | Inner Rotations (\(N_{\text{in}}\)) | Flat Line Turns | Physical Analogy |
|---|---|---|---|---|---|---|
| Identical Coins | \(1 : 1\) (\(R = r\)) | \(1.000\) | \(2.000\) | \(0.000\) (Degenerate) | \(1.000\) | Classic Coin Paradox |
| 2:1 Ratio | \(2 : 1\) (\(R = 2r\)) | \(2.000\) | \(3.000\) | \(1.000\) | \(2.000\) | Cardano Hypocycloid |
| 1982 SAT Question | \(3 : 1\) (\(R = 3r\)) | \(3.000\) | \(4.000\) | \(2.000\) | \(3.000\) | Famous SAT Error |
| Epicyclic 4:1 Gear | \(4 : 1\) (\(R = 4r\)) | \(4.000\) | \(5.000\) | \(3.000\) | \(4.000\) | Planetary Gear Carrier |
| Solar Orbit Analogy | \(365.25 : 1\) | \(365.250\) | \(366.250\) | \(364.250\) | \(365.250\) | Sidereal Year Days |
Authoritative answers to common questions about the Coin Rotation Paradox, the 1982 SAT math question error, epicyclic rolling, and sidereal day calculations.
