Solution : Circle A roule autour de Circle B

How many times will circle A revolve around itself in total ?

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Solution

This classic problem shows a smaller wheel (Circle A) rolling without slipping around a larger wheel (Circle B).
The question is: how many full revolutions does Circle A make when it returns to its starting point?
At first glance, you might think it’s simply the ratio of the perimeters, but an extra rotation appears!
This visual explanation will guide you step by step to understand why.

Let \(R\) the radius of the circle \(B\) and \(r\) the radius of the circle \(A\). The center of the circle \(A\) travels around a circle of radius \(R+r\), Not \(R\).

So the solution is

\[\begin{aligned} \frac{2\pi(R+r)}{2\pi(r)}&= \frac{R}{r} + 1\\\\ &= \frac{3r}{r} + 1\\\\ &= 4 \end{aligned}\]

This paradox illustrates that even a seemingly simple motion of one circle rolling around another hides a subtlety: an extra revolution occurs due to the circular path.
Understanding this phenomenon helps grasp relative motion, rolling without slipping, and rotations in space.
To explore more visual challenges like this, check out our “Solved Exercises” section on Art Of Mathematics.

Home -> Solved problems -> How many times will circle A revolve around itself in total ?

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How many times will circle A revolve around itself in total ?