That works well if the player is outside the lines. In the case mentioned above, the OP said the player had to be inside. This means the collision polygon created by the image is the opposite of the collision polygon needed. The player is inside the lines while the area outside the lines represent the collision polygon. Also, as you mentioned, more than 8 points degrades the performance, so your suggestion would not be optimal anyway. The image above would require 18 points to create.
You will get good performance and results if you use my suggestion above.
I hope this helps you get what you need and good luck with your project.