The golden rectangle has the following property: If a square is cut off from one side of the rectangle ...  
... the rectangle that remains ...  
... has the same ratio of sides as the original rectangle.  
If we call this ratio x, then a rectangle with sides 1 and x will have the correct ratio. The rectangle remaining after cutting away a square will have sides x1 and 1.  
Setting the two ratios equal leads to the equation

We can solve this equation for the Golden Mean: we find one positive root, x= (1+)/2 = 1.618033... This rectangle seems to have nothing to do with a pentagon. Nevertheless,