Expected Shadows

Any "surface element" of area $d\omega$ has an average projected area of $\frac{1}{2}d\omega$ . To arrive at the constant $\frac{1}{2}$ , note that the full area of a unit sphere is $4\pi$ , whereas the average projected area is $2\pi$ . In this case, a given surface element only "counts" half the time, as this is a convex surface and the projection of the "top" part overlaps and coincides with the projection on the "bottom" part. Thus, the expected area of the shadow is $\frac{1}{4}S$ where $S$ is the surface area of the unit truncated icosahedron, this gives approximately $18.151$