Using Pythagoras' Theorem the height of the triangle is $\sqrt((2r)^2 - r^2 ) = r\sqrt3$. So the area of the triangle is $r^2\sqrt3$. The area of ${1\over 2}$ a circle is ${1\over 2}\pi r^2$. So the area shaded is $r^2\sqrt3 - {1\over 2}\pi r^2$ and the proportion of the tessellation that is shaded is $$\frac{r^2\sqrt3 - {1\over 2}\pi r^2}{r^2\sqrt3} = 1 - \frac{\pi}{2\sqrt3}$$