The above is a mid-cross section.  

The radius of the sphere = OB = OC = R

The height of the cone = CD = h

OD = h-R

The radius of the cone is 
DB =  =  =  = 
 = 

Volume of cone = 









Put that = 0



Divide through by 

   

Divide through by h.  We aren't interested in when h=0

   

   

   

Since the radius = , where D is the diameter, 

   

   

So we have proved that the height is  the diameter of the sphere.

Edwin