Suppose I have several polygons of varying shape and size, each containing a single point. I want to compare the centrality of each point, and determine which point is more central to its polygon. I'm working in QGIS and R, though for this question I'll provide a generic example.
So far I have found the centroid of the polygon, and measured the distance between the point and the centroid. Initially I thought I could divide the distance by the area of the polygon in order to create a figure for comparison, but this is flawed and gives incorrect results. I then experimentally tried dividing the distance by the square root of the area instead - and this yielded what look like correct rankings. See the below illustration:
I'm having trouble wrapping my head around the mathematics here. Is this a valid way to compare centrality in non-uniform polygon shapes and sizes? Eventually I will need to apply it to more complex shapes like the following: