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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 Quick calculation (spoiler) | Comment 1 of 3
Let the radius of the circle be r.
Consider 1/6 of the hexagon, which is an equilateral triangle.
The altitude of the larger triangle is r.
The side of the smaller triangle is also r, which makes its altitude r/2*Sqrt(3).

The ratio of the larger to the smaller altitude is 1:Sqrt(3)/2, 
or 2:sqrt(3).

Squaring this, the ratio of the areas is 4:3.

So the larger hexagon = 4 square units.

  Posted by Steve Herman on 2016-01-07 17:17:23
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