To construct the regular hexagon, the radius length ab is used to mark the vertices of the hexagon. can this method be used to make any regular inscribed polygon, such as a square?

The correct answer is:

No, it will not work for all regular inscribed polygons.

Explanation:

If we were to inscribe a regular hexagon inside a circle using this method and then try to inscribe a square using the same method, we would see that the arcs made using the radius would not give us 4 congruent sides.  This would not work.