That's what I thought, too, but then I realized that what we were both missing is that the edges of the cutting surface are not the same as the edges of the sha

That's what I thought, too, but then I realized that what we were both missing is that the edges of the cutting surface are not the same as the edges of the shape. The vertices really do trace out a right angled square, so if you just put cutting surfaces at the vertices and inside them, it really will cut a square.