Notes
angle formed by three squares solution

Angle Formed by Three Squares

Solution by Trigonometry

With the points labelled as above, triangle $A B C$ is a right-angled triangle with $A C$ twice the length of $B C$ and so is half an equilateral triangle. This means that $G F C$ is also half an equilateral triangle, so if $x$ is the length of the side of one square, $C F$ has length $\frac{1}{2} x$ and $G F$ has length $\frac{\sqrt{3}}{2} x$. Then $E B$ has length $\frac{3}{2} x$ and $G E$ has length $\frac{\sqrt{3}}{2} x + x$. The tangent of angle $G \hat{D} E$ is therefore:

So to $3$ decimal places, angle $I \hat{H} G$ is: