The sides of the rectangle are parallel to the axes, so the fourth point must make a vertical line with (5,11) and a horizontal one with (16,-2); this means that the fourth point is (5,-2). The graph of the region inside the equation is a circle with radius 3 and center (5,-2):

Since each angle of a rectangle is \(90^{\circ}\) and the corner coincides with the center of the circle, the rectangle covers exactly a quarter of the circle. The area of the intersection is thus \(\frac{1}{4}r^2\pi=\frac14\cdot3^2\pi=\boxed{\frac94\pi}\) .