Question 161610
found the answer on the web.
take the hexagon and draw diagonals through each of the opposite corners.
this breaks the hexagon up into 6 equilateral triangles.
each of the sides of these equilateral triangles will be 12 inches long.
area of a triangle is 1/2 base * height.
we have the base.
we need the height.
take one of the equilateral triangles and drop a perpendicular from opposite side.
this becomes the height of the triangle.
the right triangles become 30-60-90 triangles.
in a 30-60-90 triangle, the sides opposite the angles are in the following ratio:
side opposite 30 degrees = .5
side opposite 60 degrees = {{{sqrt(3)/2}}}
side opposite 90 degrees = 1
height is side opposite 60 degrees.
base is side opposite 30 degrees.
since the base is (1/2)*L and the hypotenuse equals L, the height is equal to
{{{sqrt(3)/2}}}*L
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we now have enough to compute the area of the hexagon.
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area of the hexagon is 6 * the area of each equilateral triangle.
area of each equilateral triangle is 1/2 * base * height.
base = 12 
height = {{{sqrt(3)/2}}}*12
formula becomes
6 * (1/2) * 12 * {{{sqrt(3)/2}}}*12
answer is:
374.1229744.
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formula for area of the hexagon had become
{{{((3*sqrt(3))/2)*L^2}}}