Circles, angles, and edge splits
If your Edge Split modifier is set to 30° (by default), then you should never add circular shapes with less than 13 sides to them. Why? Because a 360-degree circle divided by 13 sides equals approximately 27.7° per side.
Any fewer sides and your angles are going to break the 30° mark and become sharp. So, 13 is the mathematical minimum. In practice, however, it's not an easy number to work with.
You can't cut it into halves or thirds, and each angle is a long decimal number (27.6923076...). This makes it hard to rotate something (such as a cut out) around a cylinder by "one side".
Instead, I suggest using a minimum ...