Quandaries
and Queries 

a belt is stretched around two pulleys whose centers are d units apart and whose radii are R and r respectively (obviously R+r<d). the challenge is to find the length of the belt, l as a formula in terms of R, r, and d only. 

Hi Ian, I drew a labeled diagram.
AB = BF = R, CD = r and BC = d. The belt section AD is tangent to the large pulley at A so angle DAB is a right angle. The line DE is constructed parallel to CB so DE = d. Since triangle AED is a right triangle
that is
Solve for AD. Since ED and BC are parallel angle AED and angle ABC are equal. But angle AED is the inverse tangent of ^{AD}/_{AE} . The measure of angle ABF is twice the measure of angle ABC and hence you can now find the length of the longer of the two arcs joining A and F. Angle BCD is 180 degrees minus angle ABC and hence you can also find the length of the appropriate arc on the smaller pulley. Harley 

