Take two concentric circle, one with radius [tex]r[/tex] and the other with radius [tex]R[/tex], and suppose [tex]r<R[/tex]. (Picture a dartboard with two circular sections.)
The area of the smaller circle is [tex]\pi r^2[/tex]. The area of the larger one is [tex]\pi R^2[/tex]. So the area between the larger circle and smaller circle is [tex]\pi R^2-\pi r^2=\pi(R^2-r^2)[/tex]
