Question

f a 12-sided regular polygon rotates about its center, at which angle of rotation will the image of the polygon coincide with the preimage?

Answer 1

This question boils down to finding the measure of the central angle of the polygon.

As an example, consider a square (4-sided regular polygon)

Notice that the sum of all central angles is \(360^\circ\). Each side corresponds with one of four congruent central angles, so each central angle has measure \(\dfrac{360^\circ}{4}=90^\circ\).

Rotating a square \(90^\circ\) will make its image exactly the same as its preimage (see sketch in next comment).

For a 12-sided polygon, each central is \(\dfrac{360^\circ}{12}=30^\circ\).