What is the ratio of the area of a square to the area of an equilateral triangle if they have the same perimeter?
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