Let the point on the inner circle be T. Because PQ is perpendicular to OT, OTP is a right triangle. Therefore OP2=OT2+PT2. Now, the true area of the doughnut is (π/2)(OP2-OT2)= (π/2)(OT2+PT2-OT2)=(π/2)(PQ/2)2.Amazing, huh? Of course the smart-ass answer is that because a formula must exist for this area based on the length of that tangent alone (or else the problem wouldn't have an answer). Since the radius of the inner circle does not matter, make it radius 0, and the area will stay the same or just the area of the remaining circle. In a non-authoritative setting this form of reasoning is invalid, because you can't be sure if the questioner is being honest in implying the existence of that formula.
Another problem will be posted tomorrow.