Henry Ernest Dudeney/Modern Puzzles/108 - Hexagon to Square/Solution

by : $108$

 * Hexagon to Square
 * Can you cut a regular hexagon into $5$ pieces that will fit together to form a square?

Solution

 * Dudeney-Modern-Puzzles-108-solution.png

Cut the hexagon in half along a diagonal and place the two parts together to form the parallelogram $ABCD$.

Construct $CF$ perpendicular to $DC$.

Produce $DC$ to $E$ where $CE = CF$.

Construct the semicircle $DHE$, whose center will be at $G$.

Produce $CF$ to $H$.

We have that $CH^2 = DC \cdot CE$.

We have that $DC \cdot CE$ is the area of the parallelogram $ABCD$, and therefore the original hexagon.

Hence a square whose side is $CH$ will have the same area as that hexagon.

Construct the semicircle $DJC$, whose center will be at $K$.

Construct the arc $HJ$ whose center is at $C$.

Draw $CJ$ and $DJ$.

Construct $LJ = CJ$ and the follows.

The rest explains itself.