The Hexagon And Its Star
Geometry
Level
1
Consider a regular hexagon. Extend the alternating sides to get a star.
Which is larger, the yellow area or the blue area?
Yellow area
Blue area
Equal
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We can determine the interior angles of the hexagon: 4 ⋅ 1 8 0 = 7 2 0 and 6 7 2 0 = 1 2 0 . Using this, we can conclude the interior angles of the surrounding triangles are 60 degrees: 3 6 0 − 1 2 0 ⋅ 2 = 1 2 0 , 2 1 2 0 = 6 0 . We can divide 360 by 6 to get 60, and counting there are 6 yellow triangles and 6 blue triangles. Since the blue triangles are equilateral and share one side with the yellow triangles, we know they are equal.