Question 315581
The apothem of an equilateral triangle is,
{{{Ap=(sqrt(3)/6)*S}}}
where S is the length of the side. 
The height of the triangle can be found using Pythagorean therorem,
{{{(S/2)^2+H^2=S^2}}}
{{{S^2/4+H^2=S^2}}}
{{{H^2=(3/4)S^2}}}
{{{H=(sqrt(3)/2)*S}}}
The area of the triangle is given by,
{{{A=(1/2)bh=(1/2)S(sqrt(3)/2)S=(sqrt(3)/4)S^2}}}
From the apothem formula,
{{{Ap=(sqrt(3)/6)*S}}}
{{{S=(6/sqrt(3))Ap}}}
{{{S^2=(36/3)A^2=12Ap^2}}}
So then
{{{A=(sqrt(3)/4)(12Ap^2)}}}
{{{A=(sqrt(3)/4)(12*36)}}}
{{{A=108sqrt(3)}}}