Coin Rotation Paradox
Calculate how many rotations a coin makes when rolling around another circle or polygon. The coin rotation paradox reveals that a coin rotating around an identical-sized coin makes 2 full rotations, not 1. Enter the radius ratio and discover the counterintuitive truth with this free interactive geometry tool.
About This Calculator
The coin rotation paradox (also known as the circle rotation paradox or SAT coin problem) is a counterintuitive geometry fact: when one coin rolls around another coin of the same size without slipping, it completes 2 full rotations -- not 1 as common sense suggests.
This happens because the rolling coin's center travels in a circle of radius equal to the sum of both coins' radii. For equal coins of radius R, the center's path has radius 2R, so the distance traveled is 2pi(2R) = 4piR. The rolling coin's circumference is 2piR, so it must rotate 4piR / 2piR = 2 times to cover that distance.
In general, for a rolling circle of radius Rsmall traveling around a stationary circle of radius Rbig, the number of rotations is:
Rotations = (Rbig + Rsmall) / Rsmall = (Rbig / Rsmall) + 1
Simply enter the ratio of the stationary circle's radius to the rolling circle's radius to compute the total number of rotations. A ratio of 1 (identical coins) yields 2 rotations -- the classic paradox.
Frequently Asked Questions
What is the coin rotation paradox?
The coin rotation paradox states that when one coin rolls around another coin of the same size without slipping, it makes exactly 2 full rotations by the time it returns to its starting position -- not 1 as intuition suggests. This is because the rolling coin's center travels a circular path of radius equal to the sum of both coins' radii (2R when equal), making its total travel distance 2 x (2piR) = 4piR, which is twice its own circumference of 2piR.
How do you calculate coin rotations?
For two circles, the number of rotations = (R_big + R_small) / R_small = (R_big / R_small) + 1, where R_big is the radius of the stationary circle and R_small is the radius of the rolling circle. For identical coins, R_big = R_small, so rotations = 1 + 1 = 2. For a coin rolling around a polygon, add the exterior angle contributions: rotations = 1 + (sum of exterior angles) / 360 deg.
Why does a coin make extra rotations?
The extra rotation comes from the rolling coin traveling around a circular path whose radius is larger than its own. The center of the rolling coin traces a circle of radius (R_big + R_small), so the distance its center travels is 2pi(R_big + R_small). Dividing by the rolling coin's circumference 2piR_small gives (R_big + R_small) / R_small = R_big/R_small + 1 rotations. The +1 term is the extra rotation that surprises most people.
Does the coin rotation paradox apply to polygons?
Yes. When a coin rolls around a polygon (triangle, square, hexagon, etc.), the same +1 rule applies. The coin rotates an additional full turn because it must pivot around each vertex by the exterior angle of the polygon. For any convex polygon, the sum of exterior angles is 360 deg, which adds exactly 1 extra rotation to the total.
Is the Coin Rotation Paradox Calculator free?
Yes, all calculators on Calculy are completely free to use. There are no subscriptions, hidden fees, or usage limits.
Where is the coin rotation paradox used in real life?
The coin rotation paradox appears in planetary gear systems (epicyclic gearing), orbital mechanics where a moon rotates around a planet, and mechanical design involving rolling elements. It is also a classic SAT and math contest problem that tests understanding of circular motion and reference frames.