Plot the sketch of the cross section of the two spheres intersecting each other. Drop an altitude from one of the intersecting point to the segment connecting the two centers.

The altitude is the radius of the circle formed by the points of intersection of the two spheres. The measure of the radius is,

r = \sqrt{6^2-5^2} =\sqrt{11}

The area of the circle is,

\pi r^2 = 11\pi