jmc

The two triangles make a smaller hexagon inside the large hexagon with the same center. Draw six lines from the center to each of the vertices of the small hexagon. Both triangles are now divided into $9$ congruent equilateral triangles, with the smaller hexagon region taking $\frac{6}{9}=\frac{2}{3}$ of the triangle.

The triangle is $\frac{1}{2}$ of the larger hexagon, so the smaller hexagon is $\frac{1}{2}\cdot \frac{2}{3}=\frac{1}{3}$ of the larger hexagon.

We now find the area of the large hexagon. By drawing six lines from the center to each of the vertices, we divide the hexagon into six equilateral triangles with side length $4$. The area of an equilateral triangle with side length $s$ is $\frac{s^2\cdot \sqrt{3}}{4}$, so the area of each triangle is $\frac{16\sqrt{3}}{4}=4\sqrt{3}$. Therefore, the area of the large hexagon is $24\sqrt{3}$. The area of the smaller hexagon, which is the region common to the two triangles, is $\frac{1}{3}\cdot 24\sqrt{3}=\boxed{8\sqrt{3}\text{ square inches}}$.