smc

The region is the shaded area: We can find the area of the shaded region by subtracting the pentagon from the middle third of the square. The area of the middle third of the square is $\left(\frac{1}{3}\right)(1)=\frac{1}{3}$. The pentagon can be split into a rectangle and an equilateral triangle. The base of the equilateral triangle is $\frac{1}{3}$ and the height is $\left(\frac{1}{3}\right)\left(\frac{1}{2}\right)(\sqrt{3})=\frac{\sqrt{3}}{6}$. Thus, the area is $\left(\frac{\sqrt{3}}{6}\right)\left(\frac{1}{3}\right)\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{36}$. The base of the rectangle is $\frac{1}{3}$ and the height is the height of the equilateral triangle minus the height of the smaller equilateral triangle. This is: $\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{6}=\frac{\sqrt{3}}{3}$ Therefore, the area of the shaded region is $\frac{1}{3}-\frac{\sqrt{3}}{9}-\frac{\sqrt{3}}{36}=\boxed{\frac{12-5\sqrt{3}}{36}}.$