The area of a shape in is defined as the -dimensional measure of the boundary of the point-set.
The definition of area given here might be a bit surprising for those who have not accustomed to it. For example, the area of a sphere of radius in is , and not (which is its volume). However, it is the correct generalization when considering geometric applications. One example is given by the surface area heuristic when building kd-trees for ray tracing.