I'm trying to draw a tin can, from scratch, but I'm having trouble with the lid. I want to put a lip, on the lid, but I've only managed to create the outside of the lip, not the inside (see photo). I want the inside to go where the red line is, but every method I've tried, the lines of the elipse cross over (see photo)

Something like this ? 

I did this using boolean operations - Path > division etc, then deleted the bits I did not need. The used union.

It's not geometrically perfect - the stroke thickness does hide the errors a bit.