To determine how many times a regular hexagon coincides with its preimage during a [tex]\(360^\circ\)[/tex] rotation, we need to analyze the symmetry of the hexagon.
### Step-by-Step Solution:
1. Understanding the Hexagon's Symmetry:
- A regular hexagon has 6 sides of equal length.
- Each angle between the sides in a regular hexagon is [tex]\(120^\circ\)[/tex].
- A regular hexagon is highly symmetrical and can coincide with its preimage multiple times when rotated about its center.
2. Rotational Symmetry:
- The hexagon can be rotated by certain angles and still look the same, coinciding with its original position.
- Specifically, a regular hexagon coincides with its preimage every time it is rotated by [tex]\(\frac{360}{6} = 60^\circ\)[/tex].
3. Calculating All Possible Coincidences:
- The hexagon coincides with its preimage at [tex]\(0^\circ\)[/tex] (no rotation).
- Additionally, it coincides at rotations of [tex]\(60^\circ, 120^\circ, 180^\circ, 240^\circ,\)[/tex] and [tex]\(300^\circ\)[/tex].
Thus, the angles [tex]\(0^\circ, 60^\circ, 120^\circ, 180^\circ, 240^\circ,\)[/tex] and [tex]\(300^\circ\)[/tex] are the instances where the hexagon coincides with its preimage within a full [tex]\(360^\circ\)[/tex] rotation.
4. Counting the Coincidences:
- Including the starting position, there are a total of 6 instances where the hexagon coincides with its preimage during a full [tex]\(360^\circ\)[/tex] rotation: [tex]\((0^\circ, 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ)\)[/tex].
Therefore, the hexagon coincides with its preimage 6 times during a [tex]\(360^\circ\)[/tex] rotation.
Thus, the correct answer is:
[tex]\[ \boxed{6} \][/tex]
