Question

A square is inscribed in an equilateral triangle. Find the ratio of area of
the square to that of the triangle.

Answer 1

Let 'b' be the side of the square
height of equilateral=b/2*sqrt(3)
Area=1/2*b*b/2 sqrt(3)=b^2sqrt(3)/4
area of @ side of a triangle=1/2*b*b sqrt(3)=b^2/sqrt(3)/3
area of the large triangle=sum of 3 triangles+area of the square
=b^2+b^2sqrt(3)/4+b^2sqrt(3)/3
hence the large triangle=1+sqrt(3)/4+sqrt(3)/3 times as large as the square
Ratio of the triangle to area of square=12+3sqrt(3)+4sqrt(3)/12=(12+7sqrt(3)/12
Ratio of the square to triangle is the inverse =12/(12+7sqrt(3))