Coin Rotation Paradox Calculator - Rolling and Sliding Count

A coin rotation paradox calculator answers a counter-intuitive geometry puzzle: when one coin rolls without slipping around an identical coin, the moving coin completes two full rotations, not one. The moving coin traces a path equal to one circumference yet ends up facing the original direction twice. The extra rotation comes from the path's curvature, not extra distance. The same idea generalizes to any pair of round disks: the moving coin completes 1 + Rf/Rr rotations, where Rf is the fixed coin's radius and Rr is the rotating coin's radius.

• • Counter-intuitive geometry puzzles: Verify the famous result that rolling one coin around another of the same size yields two full rotations.
• • Classroom demonstrations: Show students the difference between revolution and rotation, and let them change coin sizes to see how the count changes.
• • Tidal locking analogies: Compare the rolling result with the slippage case to understand why one face of Earth's Moon always points at Earth.
• • Disk-on-disk engineering: Estimate rotations of a rolling gear against a fixed disk for kinematic sketches.

Set up the experiment with two identical coins, identify the starting point of contact, and roll the moving coin around the stationary one. When the moving coin returns to the starting point, mark the orientation. The moving coin has rotated twice, even though the path length equals one circumference of the fixed coin.

Switch to a different pair, for example a half-dollar rolling around a quarter, and the rotation count changes predictably. The calculator below reproduces both experiments and accepts any size pair so you can extend the demonstration to gears, wheels, and other disks.

If your goal is to rotate a 2D point by a fixed angle rather than track a rolling disk, the Rotation Calculator covers the related point-rotation workflow.

The calculator converts both diameters into radii, applies the rolling formula N = 1 + Rf/Rr, and returns the rotation count plus the supporting lengths and angles. Sliding mode returns exactly one rotation, matching the case where the contact point slides around the fixed coin's circumference.

• Rf: Radius of the fixed (stationary) coin, computed as half the entered fixed diameter.
• Rr: Radius of the rotating coin, computed as half the entered rotating diameter.
• N: Number of complete 360 degree rotations the moving coin performs during one full revolution around the fixed coin.
• mode: Rolling applies the paradox formula; sliding locks the contact point so only one rotation occurs.
• unit: mm, cm, or inches. Both diameters use the same unit, so the radii cancel in the ratio.

The proof starts by marking a point P on the moving coin where the two coins first touch, then tracking it through one revolution. Rolling without slipping forces the arc on the moving coin to equal the arc on the fixed coin. Adding the arc contributed by the orbital revolution gives the total arc as theta times (Rf + Rr).

Setting the total arc equal to one full circumference 2*pi*Rr gives theta* = 2*pi*Rr / (Rf + Rr). Dividing a full revolution 2*pi by theta* yields N = 1 + Rf/Rr. For two identical coins Rf/Rr = 1, so N = 2, the classical paradox result.

According to Omni Calculator, the moving coin completes N = 1 + Rf/Rr rotations around the fixed coin, which equals exactly two for identical coins and one with slippage.

If you also need a standalone circumference for one of the coins outside the rolling context, the Circumference Calculator covers the basic C = 2*pi*r calculation.

Four ideas decide how the coin rotation paradox calculator interprets the inputs and explains the result.

These four ideas cover the geometry, the kinematics, the source of the extra rotation, and the slippage case. They are the building blocks for the worked examples and FAQs.

When the rotation count needs to be expressed as a rate rather than a count per revolution, the RPM Calculator converts revolutions per minute into rad/s and linear rim speed.

Enter the diameter of each coin, pick the unit, choose rolling or sliding, and read the rotation count with the supporting lengths.

1. 1 Enter the fixed coin diameter: Type the diameter of the stationary coin. The default 24.257 mm matches a US quarter.
2. 2 Enter the rotating coin diameter: Type the diameter of the coin that travels around the fixed coin. Identical coins reproduce the classical paradox result of two full rotations.
3. 3 Pick the diameter unit: Select millimeters, centimeters, or inches. The ratio Rf/Rr is unit-independent.
4. 4 Choose rolling or sliding mode: Rolling is the paradox case. Sliding is the tidal-locked case and always returns one rotation.
5. 5 Read the rotation count: The primary output is the full rotations. Secondary rows show path length, moving coin circumference, radius ratio, and total rotation angle in degrees.

For a classroom demo with two US quarters, leave the diameters at 24.257 mm and the mode at rolling. The calculator returns two full rotations, a path length of about 76.18 mm, and a rotation angle of 720 degrees. Switch the mode to sliding and the count drops to one.

If the same problem needs an arc length subtended by a partial central angle rather than a full revolution, the Arc Length Calculator handles arbitrary arc measurements.

The coin rotation paradox calculator gives you the rotation count, the supporting geometry, and the slippage case in one place.

• • Two-mode generalization: The same form handles rolling (no slippage) and sliding (tidal-locked) cases, so one page covers both physical regimes.
• • Any size pair: The formula N = 1 + Rf/Rr works for identical coins, half-coins, larger gears, and tiny ball bearings.
• • Unit-agnostic input: Millimeters, centimeters, and inches are accepted, and the calculator converts internally so the radius ratio is correct.
• • Supporting lengths in the result panel: The fixed coin circumference, rotating coin circumference, and radius ratio are returned with the rotation count for verification.
• • Direct angle readout: The rotation-per-revolution row reports the total spin in degrees, which makes it easy to compare with the 360 degrees on a flat surface.
• • Slippage case for tidal-locking demos: Sliding mode reproduces the tidal-locked count of one rotation, useful for physics and astronomy demonstrations.

Because the result panel shows the rotation count with both circumferences and the radius ratio, you can pick the intermediate value that belongs in the next step. A puzzle explanation only needs the count. A physics problem may want the arc length. A kinematics sketch may want the angle.

For a pure circumference or arc calculation on a single circle without the rolling interaction, the Circle Length Calculator covers the same C = 2*pi*r relationship on its own.

The coin rotation paradox formula is short, but five input choices decide what the calculator actually returns.

• • The calculator assumes ideal circular disks with no thickness. Real coins have finite thickness that affects rolling friction but not the rotation count.
• • Rolling mode assumes no slippage. Any real slip drops the count below 1 + Rf/Rr; sliding mode is the full-slip extreme.
• • Very small rotating coins produce very large rotation counts. Display precision is capped at four decimals.

According to Wolfram MathWorld, a full rotation about an axis is one complete turn of 360 degrees, while revolution describes travel around an external point - the distinction at the heart of the coin rotation paradox.

According to Wolfram MathWorld, the circumference of a circle equals 2*pi*r, so a path of length 2*pi*Rf traced around the fixed coin produces N = 1 + Rf/Rr rotations of the moving coin.

If you also need to convert the path length into another unit after the calculation, the Circle Perimeter Calculator translates a circle's perimeter between metric and imperial units.