The area of the deck is the area of the large circle minus the small circle. That’s πA2 – πB2, or π(A2 – B2).
We don’t know A or B, but we do know the length of the green line. That’s 42. So the Pythagorean theorem can tell us that B2 + 212 = A2, and hence that 212 = A2 – B2. Happily, we can plug this directly into the equation above and get our answer: The area of the deck is 212π, or 441π square feet — regardless of the size of the carousel!
Remarkably, this is true — so long as the green chord stays locked in position between them, the two circles can assume any size and the difference in their areas will remain constant. A similar principle holds in three dimensions.
In The Universal Book of Mathematics, David Darling writes, “An even more amazing fact is that if you slide a chord of fixed length around any convex shape C so that the chord’s midpoint traces out another figure D, the area between C and D doesn’t depend on what shape you started with.”
(Thanks, Dave.)
