[math-fun] Random nonempty intersection of balls

Let P be a point in Euclidean n-space and P' be a point chosen uniformly at random from the ball of radius 2 centered at P, so that the unit balls centered at P and P' have nonempty intersection. Show that the expected hypersurface area of the intersection of the balls (I guess the shape would be called a lunoid) is exactly equal to the hypersurface area of an (n-1)-sphere of radius 1/2. Is this observation new? What about the case n=2 where the lunoid is a lune? I know a proof, but it uses combinatorics in a way that I suspect is unnecessary. Jim Propp