Step 5: Construct the lining
You may be tempted to stitch them together in pairs and have three fat pie slices, but resist that. Stitch two segments together, then add a third, for half a pie. At this point the temptation is to add a fourth segment (and then fifth, etc) but don't do that either. Stitch the last three together just like the first three, so that you have two half-pies.
You can then sew the final seam, which will now be one straight line. Trust me that you do not want to try to stitch the sixth piece of a pie in place around the point of it. Even the last third is difficult - going around points is just Hard and there's no need to do that here.