The problem is interesting because some of these things have canonical
solutions, that is they are related by some factor of pi or the like.
I surmised that if you draw an inscribed regular pentagon and strike
an arc, centered on one apex, between two adjacent apeces, that this
might be the solution.  However, the correct answer is slightly smaller.
 
Equating the overlapped circle sectors and segments gives
 
 
4*SIN^2(T/4)=(pi-T+2*SIN(T))/(pi-(T/2))
 
T = angle from center of circle to ends of arc of second circle.
 
This gives a radius of R1 = 1.1587757347 * R
 
Jobst Brandt      <[log in to unmask]>