Aregular 40-sided polygon is rotated with its center of rotation at its center. what is the smallest degree of rotation needed to map the polygon back on to itself?

A regular 40-sided polygon has an interior angle equal to:
(40 - 2) (180) /40 = 171 degrees

The interior angle is also the smallest angle that is needed to  rotate the polygon and map it unto itself. So, the smallest degree of rotation needed to map the polygon back to itself is 171 degrees.