This is the technique I’ve used to sew a triangular gore into a cut opening in the fabric. The fabric with the cut opening is shown in grey, and the gore fabric in red.
Here is the fabric with the cut opening.
First you need to cut of the top of the triangular gore. The length of the cut should be approximately the same as double the seam allowance.
Next fold the triangle gore over so you can find the middle of the fabric. You then place the gore fabric so that the fold is placed where the cut in the main fabric ends, and so that the top of the gore fabric is one seam allowance above the point where the cut in the main fabric ends. This was done with the right side of the gore fabric facing the wrong side of the main fabric.
Then open up the fold in the gore fabric simultaneously as you pinch the main and gore fabrics together, so that they are now facing right sides together.
Continuing to pinch the two fabrics together pull a needle with thread through them right above the cut in the grey fabric. With right above I only mean a few millimeters.
You then sew a stitch to the left, and line up the gore fabric along the left cut side of the main fabric. The gore fabric will be protruding. The two fabric sides are supposed to be parallel, not supposed to have the cut sides on top of each other.
You then sew the two sides together. You can do the whole side by hand, or just do a few centimeters so you may check that it’s done correctly and then do the rest of the side by machine. I only did a few centimeters here before fastening the thread.
For the next step you prepare a new needle and thread. Pull the needle through the exact same point as where you started your first seam. Check on both sides that the needle goes through where you want it.
You then do the same on the right side as you did on the left side. Take one stitch to the left and line up the red and grey sides in parallel. Sewing down the side.
It should now look something like this one the wrong side of the fabric.
And when turning around the fabric, the right side should look something like this. Note that I haven’t completely sewn down the two sides of the fabric here. I only did a few centimeters by hand, so I could do the rest by machine.
After having sewn down both sides with machine and ironed the fabrics, the gore looked like this.
I will not be able to lay completely flat now, but when hanging down it as more volume and flare.