Question 772304
First main goal is find the side length of the equilateral triangle.  


SQUARE:  {{{27=w^2}}} for w being length of side of the square.
{{{w=sqrt(27)}}}
{{{w=3*sqrt(3)}}}
Perimeter is then {{{4*3*sqrt(3)}}}
Perimeter is {{{12*sqrt(3)}}}


Let t = side length of equilateral triangle.  Given is the perimeter is the same as that of the square, so, {{{3t=4w}}}
{{{3t=12*sqrt(3)}}}
{{{t=12*sqrt(3)/3}}}
{{{highlight(t=4*sqrt(3))}}}


WHAT WE KNOW ABOUT EQUILATERAL TRIANGLE

The altitude splits the triangle into two congruent right triangles of hypotenuse t and one leg of t/2.  The other leg, the altitude of the equilateral triangle, is then some a, a=sqrt(t^2-(t/2)^2);
{{{a=t*(sqrt(3)/2)}}}


AREA OF EQUILATERAL TRIANGLES:
Base times height then divided by 2;
{{{(1/2)(t/2)(a)}}}
{{{(1/2)(t/2)(t*sqrt(3)/2)}}}
{{{t^2(sqrt(3)/8)}}}


Finally substitute for value of t:
{{{((4/3)sqrt(27))(sqrt(3)/8)}}}
{{{(2^4*3*3*sqrt(3))/(9*2^3)}}}
{{{2*3*sqrt(3)}}}
{{{highlight(6*sqrt(3))}}}___________Final Answer


NOTE: Draw a picture will help.