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.
+23245 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
