A square is inscribed in an equilateral triangle as shown in the figure below.
Find the ratio of the area of the square to the area of the triangle.
Label the triangle as traingle(ABC), with A = left-bottom corner, B = right-bottom corner, and C = top corner.
Label the square as square(DEFG) with D = left-bottom corner, E = right-bottom corner, F on BC, and G on AC.
The bottom side of the triangle is now ADEB.
Let AD = 1.
Since angle(A) = 60o and triangle(ADG) is a right triangle, AG = 2 and DG = sqrt(3).
Since DG = sqrt(3), DE, EF, and FG are all equal to sqrt(3).
This means that the square DEFG has an area of sqrt(3) · sqrt(3) = 3.
The bottom of the triangle, ADEB = AD + DE + EB.
AD = 1, DE = sqrt(3), and EB = 1, so ADEB = 2 + sqrt(3).
The formula for the area of an equilateral triangle is A = sqrt(3) · s2 / 3
---> A = sqrt(3) · [ 2 + sqrt(3) ]2 / 3 = [ 7·sqrt(3) + 12 ] / 4
The ratio of the area of the square to the area of the triangle is: 3 : [ 7·sqrt(3) + 12 ] / 4