# Solution to the Rectangle on a Hexagon Puzzle +-- {.image} [[RectangleonaHexagon.png:pic]] > The area of the regular hexagon is $30$. What's the area of the rectangle? =-- ## Solution by [[Dissection]] +-- {.image} [[RectangleonaHexagonLabelled.png:pic]] =-- In the above diagram, triangles $G H F$ and $E B D$ are [[congruent]] so the rectangle has the same area as [[parallelogram]] $H B D F$. Then triangles $F C D$ and $H A B$ are [[congruent]], so the rectangle has the same area as the rectangle $H A C F$. From the properties of a [[regular hexagon]], this rectangle comprises four of the six [[equilateral triangles]] that make up the hexagon, so its area is $\frac{4}{6} \times 30 = 20$. ## Solution by [[Shearing]] Shear the rectangle parallel to $H B$ so that $G$ ends up at $F$, then shear the resulting [[parallelogram]] parallel to $H F$ so that $B$ ends up at $A$. This shows that the shaded rectangle has the same area as $H A C F$. ## Solution by [[Invariance Principle]] The shaded rectangle can be drawn with $B$ at any point between $A$ and $C$. In particular, it can be drawn with $B$ at $A$ which results in the rectangle $H A C F$.