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 Two hexagons (Posted on 2016-01-07)
Regular hexagons are inscribed in and circumscribed outside a circle.
If the smaller hexagon has an area of three square units, what is the area of the larger hexagon?

Source: Charles W. Trigg, Mathematical Quickies.

 No Solution Yet Submitted by Ady TZIDON No Rating

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Rotate the inner hexagon so its verticies meet the tangent midpoints of the outer hexagon. A set of six 30-30-120 triangles is formed, representing the difference in area.

Then divide the inner hexagon into 6 equilateral triangles.  Divide each of those triangles into three 30-30-120 triangles.  All 18 of these new triangles are congruent to the 6 triangles formed earlier.

Then the ratio of area is 24:18 = 4:3.  The larger hexagon has an area of four square units.

 Posted by Brian Smith on 2016-06-25 23:50:38

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