**The inner square's area is exactly half the outer square's area. **

The proof is really simple. Say the square has side of length *s*. Then, the radius of the inscribed
circle is *s*/2. In the inscribed square, then, a segment drawn from the center of the square to a corner
is *s*/2, and since that's the hypotenuse of a 45-45-90 right triangle, the apothem (half the length of the
side of the little square) must be

So, the length of the side is , and you can see where I'm going. The big square's area was , and the little one's is , exactly half.

For the other case, a circle inside a square inside a circle, the relationship is exactly the same: The inner
circle's area is half the outer one's!