Great job on the hypercube with slicing from one corner. Of course now we want to have the ability to slice from other directions, at least in directions perpendicular to (1,0,0,0) or (1,1,0,0), or (1,1,1,0) as well as the direction (1,1,1,1) that you have already programmed. I'm looking forward to a whole range of applets in the future.
With respect to the relationship between the area and the angle sum, that should set up a very nice demo. You start by observing that the area of a lune, between two semicircles of longitude, is proportional to the angle at either pole, so the area is to 4 pi (the entire area of the unit sphere) as the angle is to 2 pi. Thus the area of a lune is twice the angle of the lune. Now if we are given a spherical triangle we may consider it as the intersection of three lunes. In fact the three lunes will cover exactly half the sphere plus twice the area of the triangle. Thus 2 pi plus twice the area is twice the sum of the angles of the triangle, leading to the sought-for formula: Area = sum of angles minus pi. OK?
For all taxicab questions, I defer to my colleague, Dr. Reynolds, the specialist.