POW #4: Circle or Square?

Start with a unit square. Inscribe a circle inside it. Then inscribe a square inside that circle. Then inscribe a circle within that square. Then inscribe a square within that circle. Then... continue this process. At some point you will get an object whose area is less than 2^(-99.5). Is the first object with area less than 2^(-99.5) a square or a circle? What are the dimensions of this square or circle?

Return solutions ON PAPER to Laura Taalman, Burruss 127, by 9/29/04.
Make sure it is clear how you arrived at your answer.
Include your name and email address with your solution.