Want to swim?

The area of the square pool is $A_{pool} = s^2$ where $s$ is the side length of the pool. Then the area of the path is $A_{path} = (s + 4)^2 - s^2$ . Substituting these into $A_{path} = \frac{5}{4}A_{pool}$ and rearranging the equation into a quadratic gives $-5s^2 + 32s + 64 = 0$ Solving that equation for positive $s$ gives $s = 8$ , so the answer is therefore $A_{pool} = 8^2 = \boxed{64}$