Notes
triangle overlapping a hexagon solution

Solution to the Triangle Overlapping a Hexagon Puzzle

Solution by Pythagoras' Theorem and Properties of Regular Hexagons and Equilateral Triangles

Let $x$ be the length of one of the sides of the hexagon and let $y$ be the length of one of the sides of the triangle. Then using lengths in a regular hexagon, the length of $E C$ in the diagram is $\sqrt{3} x$. Applying Pythagoras' theorem to triangle $E B C$ shows that:

From regular hexagon, the area of the hexagon is $\frac{3\sqrt{3}}{2} x^2$ and from equilateral triangle, the area of the triangle is $\frac{\sqrt{3}}{4} y^2$. So $x^2 = \frac{16}{\sqrt{3}}$ and

Hence the area of the equilateral triangle is $13$.