Notes
stacked hexagons solution

Stacked Hexagons

Solution by Similarity, Area Scale Factor, and Properties of Hexagons

With the points labelled as above, let $a$ be the side length of the smaller hexagon and $b$ the side length of the larger hexagons. Triangle $B C D$ is equilateral, meaning that the combined length of $A B$ and $B D$ is the same as that of $A C$, so the length of $A$ to $B$ to $F$ is $2 b$. Triangle $E K F$ is also equilateral, so the length of $A$ to $B$ to $F$ is the same as that of $A$ to $B$ to $E$ to $K$, which is $3 a$. So $2 b = 3 a$. The area scale factor from the smaller to larger is then $\frac{9}{4}$ so the larger hexagon has area $\frac{45}{2}$.

The shaded triangle has base $I J$ and its height above this base is twice the height of the hexagon. Its area is therefore the same as that of rectangle $C H I J$, which is two thirds of the area of the larger hexagon, so has area $\frac{2}{3} \times \frac{45}{2} = 15$.