Nested Squares and Equilateral Triangles

Let side of square ABCD be $x$. Height of equilateral triangle with side $x$ is $\frac{\sqrt{3}}{2}x$. Length of diagonal $EFGH = x + 2 \left(\frac{\sqrt{3}}{2}x\right) = x(1 + \sqrt{3})$. Side of square EFGH $y = \frac{diagonal}{\sqrt{2}} = x \frac{1+\sqrt{3}}{\sqrt{2}}$. Ratio of areas $= \frac{y^2}{x^2} = \left(\frac{1+\sqrt{3}}{\sqrt{2}}\right)^2 = \frac{1 + 3 + 2\sqrt{3}}{2} = 2 + \sqrt{3}$.