Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
Real numbers are an excellent strategy in solving this problem. Say the radius of the circle is 1, and therefore its area is 
. Thus, the area of the square is also , and its side is, consequently, √.
Since the diagonal of a square is the same as the hypotenuse of a 45-45-90, the diagonal of the square is now √ times √2 , or √(2).
Be careful – the question asks for the diameter of the circle, which is 2. Thus, the answer is 2 : √(2).
The answer is A.
If you chose to make up a number for the square, your answer will look somewhat different but will still resolve to 2 : √(2).