I connected the centers of the 4 spheres on the corner of the pile to create a tetrahedron with edge length of 2d (1/2 d from each of the corner spheres, and d from the middle sphere). The height of this tetrahedron can easily be found using the equation Height = EdgeLength*sqrt(2/3) = 2d * sqrt(2/3).
Then you need to add d to that height. Why? Because there is still ½ d of sphere below the base of the 2d tetrahedron, and there is ½ d of sphere above the top sphere in the pile. So the height of the pile is:
2d*sqrt(2/3) +d = d * (2*sqrt(2/3) + 1)
Posted by nikki
on 2004-08-10 09:31:31