If the O point is perpendicular to the chord AB and the vertical foot is E, then both OEA and OEA are right triangles, and the chord is bisected by the diameter perpendicular to the chord.

Then EC=2, EA=4, and even Co and Ao are the radii of the small circle and the big circle of Co and Ao respectively.

From Pythagorean Theorem: CO2 = 2 2+OE 2

AO^2=4^2+OE^2

The above two expressions are subtracted, ao 2-co 2 = 4 2-2 2 =16-4 =12.

The area of the ring is s = π (ao 2-co 2) = 12π.