Notes
four equilateral triangles round a square solution

Four Equilateral Triangles Round a Square

Solution by Properties of an Equilateral Triangle

Let $x$ be the side length of the inner square, so that $x^2 = 12$. From the relationships between the lengths in an equilateral triangle, the height of the equilateral triangles is $\frac{\sqrt{3}}{2}$ and so the diagonal of the outer square is $x + \sqrt{3}x$. The area of the outer square is therefore:

The area of each equilateral triangle is $\frac{\sqrt{3}}{4} x^2 = 3 \sqrt{3}$ and so the area of all four is $12 \sqrt{3}$. Hence the shaded area is $24 + 12 \sqrt{3} - 12 - 12 \sqrt{3} = 12$.