The area not covered by six pointed star In a circle with radius $r$, two equi triangles overlapping each other in the form of a six pointed star touching the circumference is inscribed! What is the area that is not covered by the star?
Progress
By subtracting area of the star from area of circle , the area of the surface can be found! But how to calculate the area of the star?

The length of a side of an equilateral triangle is
$$\sqrt{r^2+r^2-2\cdot r\cdot r\cdot \cos (120^\circ)}=\sqrt 3r.$$
The distance between the center of the circle and each side of an equilateral triangle is $$\sqrt 3r\cdot \frac{\sqrt 3}{2}\cdot \frac{1}{3}=\frac 12r.$$
Hence, the length of a side of the smaller equilateral triangle, which is the 'corner' of the star, is
$$\frac 12r\cdot \frac{2}{\sqrt 3}=\frac{1}{\sqrt 3}r.$$ 
Hence, the area of the star is
$$\frac{\sqrt 3}{4}\cdot (\sqrt 3r)^2+3\times \frac{\sqrt 3}{4}\left(\frac{1}{\sqrt 3}r\right)^2=\sqrt 3r^2.$$
So, the answer is $$\pi r^2-\sqrt 3r^2=(\pi-\sqrt 3)r^2.$$
