Hint
Nick's right, it's a real bastard of a math problem.
assign coordinates, say the exact center of the lighthouse as (0,0), model the possible movement of the dog at the end of the leash in terms of trigonometric functions, and then crack open your old first year college calculus text to see the forumas to integrate to find the area under the curve!
As is frequently the case in mathematics, the answer is surprising simple and clean - but the solution is not!