Circles

Let the radius of the smaller circle  =  s

Let the radius of the bigger circle  =  b

And let's label the point of tangency " P ".

Side  s  is drawn from the center of the circle to  P ,  so side  s  meets  AB  at a right angle.

And by the hypotenuse-leg theorem, the two triangles are congruent.  And  AB =  80  So...

AP + PB  =  80

                               And we know  AP  =  PB

PB + PB  =  80

2 * PB  =  80

PB  =  40

By the Pythagorean theorem....

40²  +  s²  =  b²

1600 + s²  =  b²

1600  =  b² - s²

area of shaded region  =  area of bigger circle - area of smaller circle

area of shaded region  =   π b²   -   π s²

area of shaded region  =  π( b² - s² )

area of shaded region  =  1600π     (sq units)  

Thank you Hectictar