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.
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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, ...
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