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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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2 Answers 2

Let the hexagon have side $2x$. Then $\frac{1}{\sqrt 3}=\tan 30^\circ=x$. Hence the hexagon has area $6x=2\sqrt 3$.

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Draw a good picture. Join the centre of the circle to the vertices of the hexagon, and to the midpoints of the sides of the hexagon. (These midpoints are the points where the circle touches the hexagon,)

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 $12$.

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