Overlapping Triangles

Geometry Level 2

Two equilateral triangles are constructed on opposite sides of a square, pointing inwards as shown.

Find the ratio of the area of the shaded region to the area of the square (to three decimal places).

2 solutions

Marta Reece
Jul 5, 2017

Angle $BDC=30^\circ$

Set side of square $\overline{DE}=1$ , then $\overline {BC}=\frac12 \tan30^\circ=\frac1{2\sqrt3}$

Side of small equilateral triangle $\overline{AB}=1-2\times\overline {BC}=1-\frac1{\sqrt3}$

Area of small equilateral triangle $\frac{\sqrt3}4\times\overline{AB}^2$

Shaded area $=2\times$ area of small equilateral triangle $\approx\boxed{1.547}$

Rab Gani
Jul 10, 2017

Let the side of the square is 2. The shaded area = diag1.diag2/2 = (2 – 2tan15)(2 – 2 tan 30)/2. The ratio = shaded area/4 = 0.155