## Concentric Circles

The circles in the figure are concentric. The chord is tangent to the inner circle and has length $12$. What is the area of the non-shaded region?

Source: NCTM Mathematics Teacher, November 2006

SOLUTION

$x^2+6^2=r^2$
$r^2-x^2=36$
area of large circle =$\pi r^2$
area of inner circle =$\pi x^2$
area of non-shaded region=area of large circle – area of inner circle
$=\pi r^2-\pi x^2$
$=\pi(r^2-x^2)$
$=\pi 36$

Answer: $36\pi$