Let the sides of the rectangle, in \mathrm{cm}, be 4x and 5x.
Then the area of the square is 4x \times 5x\;\mathrm{cm}^2 = 20x^2\;\mathrm{cm}^2. So 20x^2 = 125, that is x^2 = \frac{25}{4}. Therefore x = \pm\frac{5}{2}, but x cannot be negative so x = \frac{5}{2} and so the sides of the rectangle are 10\;\mathrm{cm} and 12.5\;\mathrm{cm}.
Hence the rectangle has perimeter 45\;\mathrm{cm}.