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Let $IJKLMN$ be a hexagon with side lengths $IJ = LM = 3,$ $JK = MN = 3,$ and $KL = NI = 3$.  Also, all the interior angles of the hexagon are equal.  Find the area of hexagon $IJKLMN$.

 Dec 24, 2023
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Since all interior angles of the hexagon are equal, each interior angle measures 6(6−2)⋅180∘​=120∘. Let P be the center of the hexagon. Then triangle IPJ is equilateral with side length 3, so its area is 43​​⋅32=493​​.

 

 

 

The hexagon consists of six such equilateral triangles, so its area is 6⋅493​​=2273​​​.

 Dec 24, 2023

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