If the perimeter of a square is equal to the perimeter of an equilateral triangle, what will be the possible ratio of their areas?

1 Answer
May 15, 2018

The ratio between the area of the square and the area of the triangle is #\frac{3sqrt(3)}{4}#

Explanation:

Perimeter of a square: #4x#
Perimeter of an equilater triangle: #3y#

So, we know #4x=3y \iff x = 3/4y#.

On the other hand, we have:

Area of a square: #x^2#
Area of an equilateral triangle: #sqrt(3)/4y^2#

In fact, the height #h# divides the triangle in two right triangles, whose catheti are #l/2# and #h# , and whose hypothenuse is #l#, from which we have

#h = sqrt(l^2-(l/2)^2) = sqrt(l^2-l^2/4) = sqrt(3/4l^2) = sqrt(3)/2l#

and finally #A = (bh)/2 = \frac{lsqrt(3)/2l}{2}=\frac{sqrt(3)l^2}{4}#

So, we have

#A_{sq}/A_{tr} = \frac{x^2}{sqrt(3)/4y^2}#

Now, remember that #x = 3/4y# to get

#\frac{x^2}{sqrt(3)/4y^2}=\frac{9/16y^2}{sqrt(3)/4y^2}=9/16\cdot 4/sqrt(3)=\frac{3sqrt(3)}{4}#