Question

Four identical coins are placed in a square. For each coin the ratio of area to circumference is the same as the ratio of circumference to area. Then find the area of the square that is not covered by the coins.

• A. \( 16(\pi - 1) \)
• B. \( 16(8 - \pi) \)
• C. \( 16(4 - \pi) \)
• D. \( 16 \left( 4 - \frac{\pi}{2} \right) \)

Hint:

When working with geometric problems involving ratios, equate the two expressions and solve algebraically to find the unknown quantities.

Answer 1

The Correct Option is C

We are given that the ratio of area to circumference is equal to the ratio of circumference to area for each coin. The area of the coin is \( A = \pi r^2 \) and the circumference is \( C = 2 \pi r \). Therefore, we have the equation: \[ \frac{A}{C} = \frac{C}{A}. \] This simplifies to: \[ \frac{\pi r^2}{2 \pi r} = \frac{2 \pi r}{\pi r^2}, \] \[ \frac{r}{2} = \frac{2}{r}, \] \[ r^2 = 4 \quad \Rightarrow \quad r = 2. \] Now, the area of the square is \( (2r)^2 = 4r^2 = 16 \), and the area covered by each coin is \( \pi r^2 = 4 \pi \). The area not covered by the coins is: \[ \text{Area of square} - 4 \times \text{Area of one coin} = 16 - 4 \times 4 \pi = 16(4 - \pi). \]