Area of shaded part

We need to know the side length of the big square. Since we know that the length of each side of the smaller squares is $6$ , we calculate the value of $h$ (see figure). By pythagorean theorem, we get $h=\sqrt{6^2-3^2}=\sqrt{27}=3\sqrt{3}$ . Now, because of symmetry, we just multiply it by $2$ . So $2h=2(3\sqrt{3})=6\sqrt{3}$ . From the diagram we can see that the side length of the big square is $6+2h+6$ . So we substitute the value of $2h$ and get $6+6\sqrt{3}+6=12+6\sqrt{3}$ . The area of the big square is the square of the side length. So we have $(12+6\sqrt{3})^2=144+144\sqrt{3}+36(3)=144+144\sqrt{3}+108=252+144\sqrt{3}$ .

Now, we need to compute for the area of the six squares, and that is $6(6^2)=6(36)=216$ .

We also need to calculate the area of the regular hexagon. A regular hexagon is composed of $6$ equilateral triangles. All we have to do is to get the area of one regular hexagon then multiply it by six. We can used the derived formula $A=\dfrac{\sqrt{3}}{4}x^2$ where $x$ is the side length of the equilateral triangle. So the area of the regular hexagon is $6\left(\dfrac{\sqrt{3}}{4}\right)(6^2)=54\sqrt{3}$ .

Now to get the area of the shaded part, we simply subtract the areas of the six small squares and regular hexagon from the the area of the big square. We have

$252+144\sqrt{3}-54\sqrt{3}-216=$ $\boxed{36+90\sqrt{3}}$