For a fan you take the stringers out on a radii.
For Octagon you frame the cantilever at the angles & take the stringers down from that.
You don't cut the stringers to make angles. You cut the risers to match the angles and place the treads over the top of the runs.
Use a stringer stock wide enough for the back-cut of the rise & run, you typically want a min. of 3 1/2" left, (this depends on a lot of variables- for a example a short total run). If you do cut deep like that add an additional 2X4 sistered to the stringer, and shorten their on center.
If you cut a stringer to make an angle as you suggest, you have to support it from below then start the stair again.
Refer to a structural engineer so you'll do this safely.
Alter Eagle Construction & Design
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