If a square and an equilateral triangle have equal perimeters, what is the ratio of the area of the triangle to the area of the square?

1 Answer
Dec 1, 2015

#S_Delta/S_square = (4sqrt(3))/9#

Explanation:

Considering their equal perimeter, a side of an equilateral triangle #t# equals to #4/3# of a side of a square #s# because
#Perimeter = 3*t =4*s#

Therefore,
#t = 4/3s#

The altitude of an equilateral triangle with a side #t# equals to #tsqrt(3)/2#.
The area of an equilateral triangle with a side #t# is
#S_Delta = 1/2*t*tsqrt(3)/2 =t^2sqrt(3)/4#

The area of a square with a side #s# is
#S_square = s^2#

Their ratio is:
#S_Delta/S_square = (t^2sqrt(3))/(4s^2)#

Substituting #t = 4/3s#, we get
#S_Delta/S_square = (16s^2sqrt(3))/(9*4s^2) = (4sqrt(3))/9#