I'm pretty sure this one will require a MUCH better description, or a drawing.
A point 'O' is situated on a circle of radius R. With centre O, another circle of radius 3R/2 is described. Inside the crescent shaped area intercepted between these circles, a circle of radius R/8 is placed. If the same circle moves in contact with the original circle of radius R, find the length of the arc described by its centre in moving from one extreme position to the other.
I found the answer using earboth's shape, the answer appears to be . I also checked fardeen's answer, but it doesn't agree with the numerical approximation.
So, using the answer, I found the question to be,
The question is trivial after you find the angle in this shape:
And you can easily find it using law of ..... =)