# Circle rolling on a circle [duplicate]

Consider a circle with radius 3. Rolling around this circle is another circle, with radius 1. How many times has the little circle made a full turn when it has returned to it´s starting point?

## 1 Answer

The correct answer is actually...

4 times.

There's a video that demonstrates it far better than I could, but here's the synopsis:

This was an SAT question that everyone got wrong. The supposed answer was 3, following the math of $\frac{2\pi{(3)}}{2\pi{(1)}}$, which would yield 3. But as the video shows, after just one quarter rotation, the rotating circle has completed one turn. It takes a full four turns for it to complete. In fact, the formula used is: When radius A is $1/n$ radius B, the number of rotations will be $n+1$.

See the video here.

• Interesting. After going over the simulation in my head, it does make sense. – SMS von der Tann May 7 '16 at 18:25
• rolling around the inside of the circle it would be only 2 times, for much the same reason, – Jasen May 8 '16 at 4:50