What is the area of a hexagon with an apothem of 4 square root 3 ?

A hexagon is six equilateral triangles; the apothem is the height of one of them.

The right triangle associated with that apothem has sides in ratio so the hypotenuse, which is the side of the equilateral triangle, is

so an area of a single triangle

The entire hexagon is six of these

let m = monomial

let t = trinomial

so, m times t = t.

monomial times a trinomial = another trinomial.

geometry: triangles

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call the larger triangle abc

then the smaller triangle cde is similar to abc (all angles are equal, aaa).  

ab is the side that is 36, and we wish to find de, the side corresponding to ab

area_cde = area_abc - area_cde --> area_abc / area_cde = 2

let x=length of de

ratio of areas = (ratio of sides)^2

+2+=+%2836%2fx%29%5e2+

+2+=+1296%2fx%5e2+

++x%5e2+=+1296%2f2+=+648+

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