smc

Let point $A$ be a vertex of the regular hexagon, let point $B$ be the midpoint of the line connecting point $A$ and a neighboring vertex, and let point $C$ be the second intersection of the two semicircles that pass through point $A$. Then, $BC=1$, since $B$ is the center of the semicircle with radius $1$ that $C$ lies on, $AB=1$, since $B$ is the center of the semicircle with radius $1$ that $A$ lies on, and $\angle BAC=60^{\circ }$, as a regular hexagon has angles of 120${}^{\circ }$, and $\angle BAC$ is half of any angle in this hexagon. Now, using the Law of Sines, $\frac{1}{\sin \angle ACB}=\frac{1}{\sin 60^{\circ }}$, so $\angle ACB=60^{\circ }$. Since the angles in a triangle sum to 180${}^{\circ }$, $\angle ABC$ is also 60${}^{\circ }$. Therefore, $△ABC$ is an equilateral triangle with side lengths of $1$. Since the area of a regular hexagon can be found with the formula $\frac{3\sqrt{3}{s}^2}{2}$, where $s$ is the side length of the hexagon, the area of this hexagon is $\frac{3\sqrt{3}(2^2)}{2}=6\sqrt{3}$. Since the area of an equilateral triangle can be found with the formula $\frac{\sqrt{3}}{4}{s}^2$, where $s$ is the side length of the equilateral triangle, the area of an equilateral triangle with side lengths of $1$ is $\frac{\sqrt{3}}{4}\left(1^2\right)=\frac{\sqrt{3}}{4}$. Since the area of a circle can be found with the formula $\pi r^2$, the area of a sixth of a circle with radius $1$ is $\frac{\pi (1^2)}{6}=\frac{\pi }{6}$. In each sixth of the hexagon, there are two equilateral triangles colored white, each with an area of $\frac{\sqrt{3}}{4}$, and one-sixth of a circle with radius $1$ colored white, with an area of $\frac{\pi }{6}$. The rest of the sixth is colored gray. Therefore, the total area that is colored white in each sixth of the hexagon is $2\left(\frac{\sqrt{3}}{4}\right)+\frac{\pi }{6}$, which equals $\frac{\sqrt{3}}{2}+\frac{\pi }{6}$, and the total area colored white is $6\left(\frac{\sqrt{3}}{2}+\frac{\pi }{6}\right)$, which equals $3\sqrt{3}+\pi$. Since the area colored gray equals the total area of the hexagon minus the area colored white, the area colored gray is $6\sqrt{3}-(3\sqrt{3}+\pi )$, which equals $\boxed{3\sqrt{3}-\pi }$.