# Number of square units in area of regular hexagon with unit circle inscribed

A unit circle is inscribed in a regular hexagon. What is the number of square units in the area of the hexagon in the simplest radical form?

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This divides the hexagon into $12$ congruent triangles, each of which is a "$30$-$60$-$90$" triangle.
Take one of these triangles. If the radius of the circle is $r$, then one of the legs of the triangle is $r$. Using facts about $30$-$60$-$90$ triangles, or information about the exact tan of $30^\circ$, you should be able to find the other leg of the triangle, and hence its area. Then multiply by $12$.
Let the hexagon have side $2x$. Then $\frac{1}{\sqrt 3}=\tan 30^\circ=x$. Hence the hexagon has area $6x=2\sqrt 3$.