Hi, this is another question from the article "The Mathematics of Doodling" in the Mathematical Monthly of February 2011. (See http://math.stanford.edu/~vakil/files/monthly116-129-vakil.pdf)

On pp 126, the article mentions a remarkable fact:

Theorem 3. The average length of the shadow of a convex region of the plane, multiplied by $\pi$, is the perimeter.

Followed by:

Theorem 4. Consider the average area of the shadow of a convex region of three-space, and multiply by 4. The result is the surface area.

Are these well-known facts? I haven't heard of these facts before! Any ideas on how one could prove these two results?

Thank you in advance.