[math-fun] Random slice of a cube

If I choose a random plane that intersects a cube, creating a polygon, how many sides does the polygon have on average? (The most lowbrow way to define the probability measure I have in mind when I say "random plane" is to circumscribe a sphere of radius r around the cube with center O. Choose a point uniformly P at random on the sphere, and now choose a point Q uniformly at random on segment OP. If Q is in the cube, erect a plane through Q perpendicular to OP. Otherwise, start over. The most highbrow way is to go via the standard measure on the affine Grassmannian of all planes in 3-space; if we restrict to the set of planes that pass through the cube, we get a finite measure, so we can rescale to get a probability measure.) Jim Propp