Aregular hexagon rotates counterclockwise about its center. it turns through angles greater than 0 degrees and less than or equal to 360 degrees. at how many different angles will the hexagon map onto itself?

1. Check the pictures.

2. A regular hexagon is a very beautiful geometric shape. It is made up of 6 equilateral triangles. So many properties can be noticed from this fact. 

For example: m(AOB)=m(BOC)=...=m(DOA)=60°                          (picture 1)

3. So once you rotate the hexagon counterclockwise 60°, "click", A goes to B, another 60, A goes to C, and so on, after 6-60°, A is back to A.     (picture 2)

4. The hexagon will map to itself after 60° (A goes to B), 120° (A goes to C), 180°, 240°, 300° and 360°. So at 6 angles.