For some people, it is a mystery how to measure out the hole you need to cut in a carpet for the spindles or balusters on a stairway. Actually the key is to not measure at all. I recommend that you first practice with a piece of paper. Of course the real rug will be harder to deal with, because it is thicker and you have twenty more feet of it to contend with as you work your way down the stairs, but once you get the hang of doing it with paper, you will be ready for a real rug.
Tape a piece of paper to one of the steps so it will stay put. Have the paper flowing up the stair spindle, as would a rug intended to wrap right over the edge of the stair tread.
Push it tightly into the joint between the tread and spindle making sure that the pattern of the rug threads that are running across the tread now run straight up the spindle. If it is twisted to one side, you will end up with problems.
Slice right along the spindle. I prefer to start at the bottom and slice up as in the first photo; some people prefer going from the top down. The little straight piece between the spindle and the riser will now fall into place.
Now cut across that joint between the spindle and the tread as in the second photo. That will cause the rest of the rug to lie down across the tread with a little flap the width of the spindle standing up as in the third photo. Now cut at the joint between the tread and the spindle again just to the back of the spindle.
That will allow you to lie the rug down behind the spindle and cut off that last little flap. The carpet can now be wrapped around the end of the tread as in the last photo and when done properly, you cannot see the first slice that gave you access to cut the hole out for the base of the spindle.
I would recommend that you work with a helper whose job will be to hold on to the excess rug so that it doesn't pull your work off to one side.