>US aircraft carrier planes have a ~500 mi effective radius without refueling; even if you see a plane, all you know is that the ship might be in a ~3,142 square mile area.
pi*(500 miles)^2 = 785,400 sq. miles.
Of course I meant, 'within a circle of 3,142 mi circumference'. But no I didn't - how embarassing. I leapt at thinking '1,000 x pi is the operating area of an aircraft carrier - so perfect.'. 785,400 sq miles is more impressive and harder to find.
That explains the downvotes!