This is a cross section cut through the center of the sphere:



The lateral area of circular cylinder is

      

      

      

By the Pythagorean theorem (refer to the drawing): , so we substitute:

      

Since square roots are difficult to work with, let's square both sides:

      

The trick here is that if we maximize the SQUARE of the lateral area,
we will also have maximized the lateral area.  So we let S = A²

      

      

      

We set that equal to zero:

      0 

      0

Divide through by constant 

      0

      r=0;         R² - 2r² = 0
(min, area = 0)        -2r² = -R²
                         r² = 
                          r =  

So the radius of the cylinder which has maximum
surface area is 

       Since h = 
             h = 
             h = 
             h = 

   height = 2h = 

Edwin