Coin Rotation Paradox Calculator

The coin rotation paradox is a classic geometry result that surprises people the first time they see it. Imagine one coin rolling around the outside of another coin without slipping. If both coins have the same radius, many people guess that the moving coin should make exactly one full turn because the coins appear to have the same size. In reality, the moving coin completes two full rotations by the time it returns to its starting point. That extra turn is the "paradox". It is not a contradiction in mathematics; it is a contradiction between intuition and the actual geometry of rolling motion. The key idea is that the moving coin is doing two things at once. First, it is rotating because its edge is rolling along the boundary of the fixed coin. Second, its center is orbiting around the fixed coin's center. When we focus only on edge contact, we tend to imagine the motion as if the coin were rolling in a straight line. But the path is not straight. The center of the moving coin traces a circle whose radius is the sum of the two coin radii, R₁ + R₂. That orbital path changes the orientation of the moving coin as it goes around, and that orientation change contributes the extra rotation people often forget. For a moving coin of radius R₁ rolling around a fixed coin of radius R₂, the exact number of rotations is (R₁ + R₂) / R₁. When the radii are equal, the formula becomes (R + R) / R = 2, which explains the famous equal-coin case. If the moving coin is smaller than the fixed one, the rotation count grows large because the small coin must spin many times to traverse the relatively longer path around the big coin. If the moving coin is larger than the fixed one, the count is less than two, since the big coin covers its own circumference quickly relative to the short fixed-coin perimeter. The same formula works smoothly for fractional radii as well, which makes it useful for classroom demonstrations, puzzle explanations, and geometry explorations. This calculator gives the exact decimal result instantly. It is helpful for students learning about rolling without slipping, teachers explaining why intuition can fail in circular motion, and anyone exploring elegant mathematical paradoxes. By showing the formula directly, the tool makes it clear that the surprising extra rotation comes from orbital geometry, not from a hidden trick.