# Hexagon in a Star in a Hexagon +-- {.image} [[HexagoninaStarinaHexagon.png:pic]] > Corresponding sides of the two regular hexagons are parallel. If the small hexagon has an area of $18$, what's the shaded area? =-- ## Solution by [[Similar Triangles]] and [[Crossed Trapezium]] +-- {.image} [[HexagoninaStarinaHexagonLabelled.png:pic]] =-- In the above diagram, the line segments $A B$ and $D E$ are [[parallel]]. Angles $E \hat{C} D$ and $A \hat{C} B$ are [[vertically opposite]] so are equal. Put together, this means that triangles $D C E$ and $A C B$ are [[similar]]. Triangle $E C F$ is [[equilateral]] and $B$ is the [[midpoint]] of $C F$, so the length of $C B$ is half of that of $C E$, which has equal length with $C D$. This means that the scale factor from triangle $A C B$ to $D C E$ is $2$. So $D E$ has twice the length of $A B$, and since the [[area scale factor]] is the square of the length scale factor, the area of the outer hexagon is $4 \times 18 = 72$. The quadrilateral $A B E D$ is a [[trapezium]] since $A B$ and $D E$ are [[parallel]]. Its area is one sixth of the difference between the hexagons, so has area $9$. By considering the areas in a [[crossed trapezium]], triangles $D C A$ and $E C B$ have twice the area of $A C B$, and $D C E$ has four times that area. Putting that together, triangle $A C B$ has area one ninth of the whole trapezium, so the shaded region inside the trapezium has area $5$. The entire shaded region therefore has area $30$.