# Equilateral hexagon and a Circle

In the following diagram $ABCDEF$ is a equilateral regular hexagon with $AB = 1$
A circle is drown with radius $2$ with point $E$ as a center. What is the area of the shaded region of the circle which is outside of the hexagon.

The figure is misleading. Notice that for a regular hexagon (assuming that was what you meant by "equiangular"), the length of the main diagonal is twice the side length, so $BE = 2$ and thus the entire hexagon $ABCDEF$ lies within the boundaries of circle $E.$
As the area of the circle is $A_{\text{cir}} = \pi r^{2} = 4\pi$ and that of the hexagon is $A_{\text{hex}} = \frac{3\sqrt{3}}{2} \cdot s^{2} = \frac{3\sqrt{3}}{2},$ your answer is simply $\boxed{4\pi - \frac{3\sqrt{3}}{2}}.$