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 Ö3 divided by 2

So the dimensions of each rectangle are 1 and Ö3

The side of the square hole is Ö3 - 1 and the hole's area is the square of that :

4 - 2Ö3