Question 876802
Examining this part:  (The figure is a circle, with radius r, with another shaded circle surrounding it. The shaded part has a width marked 'w'.) 


The shaded region would be surrounding the unshaded smaller circle.  You want the area for the shaded region.  This will be a difference.  


Larger circle area, {{{pi*(r+w)^2}}}.
Smaller circle area, {{{pi*r^2}}}.
The shaded region should be the larger region minus the small circle region, leaving just the circular shaded region with the width of w.


{{{A=pi(r+w)^2-pi*r^2}}}
{{{pi(r^2+2rw+w^2)-pi*r^2}}}
{{{pi(r^2+2rw+w^2-r^2)}}}
{{{pi(2rw+w^2+r^2-r^2)}}}
{{{pi(2rw+w^2+0)}}}
{{{pi*2*rw+pi*w^2}}}-----which should be clear enough.
{{{A=2pi*rw+pir^2}}}---commutative property for multiplication.