So, this is a little tricky kind of a question and I'm not totally sure if it's a mathematic question or a more programming one, but I nevertheless hope to find answers.
I want to find out the error of a volume and surface calculation of a 3-D sampled sphere. That means, the sphere is lying in a 3-D grid and is downsampled to/by that grid. That means each voxel is either one or zero, depending on how much of the sphere is touching that voxel (to make it easy, let's assume 50%). The question now is, how much the Volume of the sphere differs from that of the cubes.
Another way of asking would be: How many equal sized cubes that share one surface can be placed in a sphere?