LHIRT POD #2 solution

Solution for POD #2 below (white on white below):

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.

LHIRT's POD #2

The problem derives itself from an old puzzle book that solves this problem in a very clever way. I am looking for the real solution, which isn't more difficult than the original method to solve it.

Take a pair of concentric circles, that share a center O. Take a point on the inner circle, and extend its tangent to its intersection with the outer circle at points P and Q. Given only the length of this chord PQ of the outer circle, calculate the area between the two concentric circles.

I will give both solutions in the morning, but I am looking for the one that does not assume that a formula exists in the first place.