find area of darkened figure 
We know that length of diameter $AB=6$; $CA=2\sqrt{3}$; 
 We should find area of darkened figure I have tried  following :
Since $AB$ is diameter  we  can say $ACB$ is right triangle  so 

we can caluclate  area of $ABC$, but my question is how calculate   area  of this triangle inside circle? Please help me.

From the metric data you can compute the angle at $B$ since $\tan B = \sqrt 3/3$.
Let $D$ be the intersection of $BC$ with the circle. Since the angle at $B$ is known  and $OB=OD$, you can easily compute the area of the circular sector $AOD$ and of the triangle $BOD$.
