Jim Propp> One tends to think that the random variable associated with the larger circle would have more variance. But as many of you have probably figured out by now, the two random variables have the exact same distribution. A more symmetrical way to describe this random variable is as the length of the vector u-v (or if you prefer u+v), where u and v are plane vectors of length 1 and 2, respectively, chosen uniformly and independently. If you exploit symmetry by choosing u to be a fixed vector of length 1 and letting v vary over all vectors of length 2, you get one of the two asymmetric descriptions; if you instead choose v to be a fixed vector of length 2 and let u vary over all vectors of length 1, you get the other one. Jim Propp It's easier than that. Uniform on the circles means uniform angles wrt x-axix, e.g. The segment joining (1,0) to 2(cos t, sin t) is obviously congruent to the one joining (2,0) to (cos t, sin t), length = sqrt(5 - 4 cos t). Interesting: The expected length of the third side of SAS triangle x,t,y for t uniform in (0,π) is (2*(x + y)*EllipticE[(4*x*y)/(x + y)^2])/Pi . So (1/2)*(x + y)*EllipticE[(4*x*y)/(x + y)^2] is an interesting sort of mean. --rwg