As shown in the figure below, four identical coins are placed in a square. For each coin, the ratio of area to circumference is same as the ratio of circumference to area. Find the area of the square that is not covered by the coins.

The correct answer is (C) 16(4 - $\pi$).

• Given: For each coin, Ratio of Area (A) to Circumference (C) = Ratio of Circumference (C) to Area (A), i.e., A/C = C/A. Since A = $\pi$ r^2 and C = 2$\pi$ r, the equation becomes $\pi$ r^2 / 2$\pi$ r = 2$\pi$ r / $\pi$ r^2, which simplifies to r/2 = 2/r, resulting in r^2 = 4 and r = 2.
• Step-by-step: The four identical coins are placed in a square. Since each coin has a radius of 2, the diameter of each coin is 4. Placing four such coins in a square implies the square's side length is twice the diameter, 4 + 4 = 8. Area of the square = 8^2 = 64. Total area of four coins = 4 * ($\pi$ r^2) = 4 * ($\pi$ * 2^2) = 16$\pi$.
• Result: The area not covered by the coins = Area of square - Area of four coins = 64 - 16$\pi$ = 16(4 - $\pi$). Other options do not follow the geometric derivation of the square's side length relative to the radii of the four coins.