The ratio does not change substantially with the dimension of the image. I don't know why you say it does. It is a dimensionless quantity. As long as you have a decent number of pixels it will be the same for a 1 megapixel image, or a 20 megapixel image. Sure it may be different if your triangle or hexagon is only 5 or 10 pixels big, because the shape is so quantized with such a small shape, but as long as you have a good number of pixels, the triangle should give you a higher ratio, like 2 or more. The more circular it gets (like a hexagon is more circular than a triangle), the close to 1 it becomes.
In your earlier post I think I gave you an answer to look in the FAQ for code to find "kinks" in the boundary. You can still use that if you want.