Relating arc lengths between a circle within a larger circle (no-slip)...

Hi, I recently had a vibrations problem in my college course that dealt with a small "pipe" inside a larger pipe. The pipe would roll within the larger pipe with no slip (see page 1 and page 2 of the solution).

Thus, a relationship could be "derived," if you will, between the arc lengths of the two surfaces.

I thought this was a simple ratio relationship (like gear ratios), in which the small pipe's rotation about the center of the large pipe would be and the small pipe's rotation about its own center would be for "small pipe."

The relationship I found was that , where is the small pipe's radius and is the large pipe's.

However! The book's solution (the one in the links) introduces a third angle and ends up saying that . Meaning, the arc length between the two pipes is related by whereas before it was

Could anyone tell me why this is and what is wrong with assuming that there is a direct relationship between the product of both radii and angles?

Thanks!