Notes
overlapping triangle and hexagon solution

Solution to the Overlapping Triangle and Hexagon Puzzle

Solution by Properties of Regular Hexagons and Equilateral Triangles

Let $x$ be the length of one side of the hexagon. Using lengths in a regular hexagon, the equilateral triangle has height $2 x$ and so has base $\frac{4}{\sqrt{3}} x$. Its area is therefore $\frac{4}{\sqrt{3}} x^2$. The two remaining parts of the hexagon each can be reassembled into an equilateral triangle with side length $x$, so each has area $\frac{\sqrt{3}}{4} x^2$. The total area is therefore:

The unshaded area consists of the hexagon without the two upper triangles, so has area equal to four equilateral triangles with side length $x$. Its area is therefore $\sqrt{3} x^2$. The shaded area is therefore

The fraction that is shaded is then $\frac{5}{11}$.