2013 AMC 10B Problem 7

Six equally spaced points on the circle form a regular hexagon. A triangle using three of them is not equilateral or isosceles only when its angles are $30^\circ,60^\circ,90^\circ$. The hypotenuse is a diameter of the unit circle, so it has length $2$. The legs of the $30$-$60$-$90$ triangle are $1$ and $\sqrt3$. The area is $\frac12\cdot1\cdot\sqrt3=\frac{\sqrt3}{2}$. Thus, the correct answer is B .