Belt

Herbert of Sha Tin College, Hong Kong submitted the only correct solution to date which is given below. Can anyone give an alternative solution?

$\alpha = \sin^{-1}(R - r)/(R + r)$ $L_1 = R(\pi + 2\alpha)$ $L_4 = r(\pi - 2\alpha)$ $L_2 = L_3 = x$ $x^2 = (R + r)^2 - (R - r)^2$ $x^2 = 4Rr$ $x = 2\sqrt{Rr}$ $L_2 + L_3 = 4\sqrt{Rr}$

The total length $L$ is $4\sqrt{Rr} + \pi(R + r) + 2(R -r)\sin^{-1}(R - r)/(R + r).$