I found this page with instructions to create a pentagon from a square paper:
Fold the square in half to create a rectangle
Mark half in the right side:
- Mark half in the down side:
- Continue folding following the diagrams:
- Fold to create a 90° angle and cut over this line:
And you have a regular pentagon:
Folding from the bottom mark and matching the right mark with the top line seems to be the most important part, by creating the angles needed for the construction. This fold reminds me the origami trisection.
I think this is not an approximation but a perfect regular pentagon. How can be proved that the angles in this construction are those from a regular pentagon?