[![The figure shows a hexagon and triangle tiled by six identical tiles.][1]][1]

  [1]: https://i.stack.imgur.com/tmEiE.png

My question is:

What is the *smallest* number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such away that all edges of the tiles are all parallel to an edge of the tiled figure?

Small print: 

 - I define a polygonal as a figure comprised of a finite set P of at least three points, together the lines x permuation $\pi$