Design a tangram

Question

Is there an hexagon with area a perfect square? If so how can I cut the square to rearrange it into an hexagon? Is it possible to do this?

If there is no hexagon with area a perfect square at least there are some hexagons with area a sum of perfect squares. For instance an hexagon with sides of length $\ell=3$ and distance from the vertices to the center $r=5$. According to my calculations (which may be wrong) this area is equal to $3\cdot 6\cdot 4=72=6^2+6^2.$

My goal is to design a tangram so I can transform a square or a set of squares into an hexagon.

General results and answers for other regular polygons are also welcome.

Answer 1

Here there is an example of a possible dissection of an hexagon in 5 pieces which form a square. The sides are not integer numbers, but this should not be a problem since it is a geometrical problem, not numerical.

Wolfram has also some general results about dissections, together with a large biography.

Answer 2

First of all, your proposed hexagon is not regular. The lines from to neighboured vertices to the center and the side between those vertices always for an equilateral triangle, as the angle at the center is $60^{\circ}$ and both sides going from the center have the same length. Therefore the distance between vertex and center is always equal to the side length. That means $A=\frac{3\sqrt{3}}{2}l.$

The $\sqrt{3}$ in there should obviously tell you, that this is never going to be a natural number for $l \in \mathbb{N}$ while a square with whole numbered side length will always have a whole number as it's area as well.