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Two hexagons (Posted on 2016-01-07) Difficulty: 3 of 5
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    
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Solution Simple Solution Comment 3 of 3 |
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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