Notes
two rectangles in a hexagon ii solution

Solution to the Two Hexagons in a Rectangle II Puzzle

Solution by Dissection, Angles in a Regular Hexagon, and Properties of Equilateral Triangles

With the diagram labelled as above, triangle $A B I$ is an equilateral triangle as it is formed by extending the sides of a regular hexagon. Therefore, triangle $A D G$ has angles $60^\circ$, $90^\circ$ and $30^\circ$. The orange lines therefore dissect the triangle into equilateral and half equilateral triangles. Counting the triangles, the hexagon consists of six equilateral triangles, and the dark blue shaded region comprises four equilateral and five half equilateral triangles. This is repeated in the upper half of the rectangle. So the total shaded region consists of thirteen equilateral triangles out of a total of twenty-five, meaning that $\frac{13}{25}$ of the rectangle is shaded.