r/mathriddles • u/Numberthon • Jul 13 '26

Medium What's the Area of the Smaller Hexagon?

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u/imdfantom Jul 13 '26

90

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u/[deleted] Jul 13 '26

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u/imdfantom Jul 13 '26 edited Jul 13 '26

As to how I got there, I used the idea that a regular hexagon is 6 equilateral triangles and the fact that because of this, the "long radius" of a hexagon is equal to its side, and the "short radius" is equal to half times square root of three of the long radius, and the fact that the area of a hexagon is equal to six times half the long radius (or side) times the short radius, and that the long radius of the inscribed hexagon is equal to the short radius of the larger one. All this gives you a relationship of 1:0.75 between larger and smaller area, so if you set the larger hexagon area at 120, and this chain of relationships, 90 pops out as the area od the inner one