The following is a variant of a problem that appeared on a college entrance examination. In the figure below, the radius of circle A is 2 units, the radius of circle B is 3 units. Starting from the position shown in the figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A first reach its starting point?

Find the general formula for circles of any radius. That is, let circle A have radius r, and let circle B have radius R. Find the number of revolutions in terms of r and R that circle A will make as it rolls around circle B.

The original problem on the college examination had circle A with radius 1, and the correct answer was not among the choices! (It was a multiple choice standardized test.)

Solution:
Since the arc length on a circle is given by s = rq, then after circle A rolls some around circle B we should have 3q = 2j. Also, after circle A does one full revolution, the angle it has rotated, j, is 2p – q. We can see this because the radii form a line, and q makes alternate interior angles with two (parallel) horizontal lines.

Hence 3q = 2(2p – q) so 5q = 4p, or q = 4p/5 (for circle A to make one complete revolution).
The number of revolutions circle A will make is 2p/q = 2.5.

In general, if circle B has radius R and circle A has radius r, then Rq = rj, and we still have j = 2p – q. So Rq = r(2p – q), which gives (R/r + 1)q = 2p, so 2p/q = 1 + R/r. That is, the number of revolutions circle A will make is n = 1 + R/r.

If the circumference of circle B was flat, then circle A would make R/r revolutions. But because it goes around circle B, there is one extra revolution. Try this with two coins of the same size. Also note that as circle B's radius approaches 0, the number of revolutions approaches 1. (Circle A rotates around a point.)