2003 AIME I Problems/Problem 5
Problem
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is 
where 
and 
are positive integers, and 
and are relatively prime, find 
Solution
The set can be broken into several parts: the big parallelepiped, the 6 external parallelepipeds, the 1/8-th spheres (one centered at each vertex), and the 1/4-th cylinders connecting each adjacent pair of spheres.
The volume of the parallelepiped is 
cubic units.
The volume of the external parallelepipeds is 
.
There are 8 1/8-th spheres, each of radius 1. Together, their volume is 
.
There are 12 1/4-th cylinders, so 3 complete cylinders can be formed. Their volumes are 
, 
, and 
.
The combined volume of these parts is 
.
