Hi Mark, yeah like Steve writes above probably your best bet is to make a 2D curve as the cutting object, and have the ends stick out a little instead of trying to make something that directly hugs the outer surface. It's fairly easy when making things just skim along each other that they might not quite fully cross the entire piece, by having things stick out a bit it eliminates any problems related to that.

I've attached a 3DM file with a version set up like that - this is ready to boolean, select the ring solid, run Construct > Boolean > Difference, and then pick the 4 curves as the cutting objects and it will divide it up into 4 pieces.

When possible it can be good to use 2D curves as cutting objects rather than messing around with constructing cutting surface pieces, it just saves some steps.

- Michael

Stevecim is right but you need to use a boolean difference to cut the band. not the merge

I find that if you first join the cutting lines check they are on the same plane (ie thickness 0) then extrude them through the object. this often gives a better cut, also you can visually check the it goes right outside the object to be cut.

Your posted 3dm has the parts chopped into 12 or more parts (was this from the aborted cutting)
cheers
shayne