Incidentally, did you notice that the yellow and purple triangles have the same area ? This doesn't require the particular case of one triangle being equilateral, any rectangle split into 4 areas by its diagonals will do.

More obvious now ?

Anyway back to the area of the Square Hole :

Thank-you to Clem, and to Marta & Brittany from MaST Community Charter School, and others who sent in solutions.

Seeing the image as a 'hole' surrounded by four rectangles, with each rectangle made from $2$ yellow (equilateral) and $2$ purple triangles.

The 'height' of the equilateral triangles is $\sqrt{3}$ divided by 2

So the dimensions of each rectangle are $1$ and $\sqrt{3}$

The side of the square hole is $\sqrt{3} - 1$ and the hole's area is the square of that :

$4 - 2\sqrt{3}$