jmc

Look at the right triangle that consists of a blue side, a red side, and a grey side. The grey side has length $3$ (half the length of the square side length). Since the red side has length $2\sqrt{3}$, by Pythagoras the blue side has length $\sqrt{3}$; thus, the right triangle is a 30-60-90 triangle, with area $\left(\frac{1}{2}\right)(3)(\sqrt{3})=\frac{3\sqrt{3}}{2}$.

The two red radii slice out a sector of 30 degrees, with area $\left(\frac{30^{\circ }}{360^{\circ }}\right)(2\sqrt{3}{)}^2\pi =\pi$.

The square consists of 8 right triangles and 4 sectors and one grey shaded area. Thus, the grey shaded area has area $$6^2-8\left(\frac{3\sqrt{3}}{2}\right)-4\pi =\boxed{36-12\sqrt{3}-4\pi }.$$