There is a square with side . Inside of this square, there are four squares on each corner. These small squares have side . There are two more small squares inside the big square, and they also have side . These two small squares touch each other by their vertices. These two squares are in the middle of the big square, and each touch one of the squares that are on the corners, creating a "chain", in such way that the diagonal of the big square can be expressed as , where is the diagonal of the small squares. After all, the big square has now a space that is not filled with small squares.
Given that the apothem of the small squares is 1 cm, calculate the area of this space. Give your answer in .
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The apothem of the small squares is 1 cm. Therefore, a/4 = 2 cm and a = 8 cm. One small square's area is 2^2 = 4 cm^2. 6 squares equal 24 cm^2. The area of the big square is 8^2 = 64 cm^2. 64 - 24 = 40 cm^2.