Let each side ot the equilateral triangle be x.
Then, its perimeter is 3x.
The formula for the area of an equilateral triangle is: Area = ( sqrt(3)/4 )·side2.
For this triangle: Area = ( sqrt(3)/4 )·x2.
Since the perimeter of the equilateral triangle is 3x, the perimeter of the square must also be 3x.
Perimeter of the square = 4·side.
For this square: 3x = 4·side, so each side = (3/4)·x.
The formula for the area of a square is: Area = side2.
For this square: Area = [ (3/4)·x ]2 = (9/16)·x2.
Ratio of the area of the equilateral triangle to the area of the square
= ( sqrt(3)/4 )·x2 / (9/16)·x2 = ( sqrt(3)/4 ) / (9/16) = 4·sqrt(3) / 9