# Stacked Hexagons +-- {.image} [[StackedHexagons.png:pic]] > Two of the regular hexagons are identical; the third has an area $10$. What's the area of the red triangle? =-- ## Solution by [[Similarity]], [[Area Scale Factor]], and [[hexagon|Properties of Hexagons]] +-- {.image} [[StackedHexagonsLabelled.png:pic]] =-- 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 $\frac{2}{3} \times \frac{45}{2} = 15$.