Let me back up and say that from what I understand there are two common ways to calculate the centroid of a polygon in ArcGIS Desktop:
- Using Calculate Geometry on fields within the attribute table of the feature class.
Data Management -> Features -> Feature to Pointfrom the toolbox.
These both give the same result - the geometric centroid of the polygon. However, there is no guarantee that point lies inside the polygon.
The Feature to Point tool has an inside checkbox option, that according to the documentation:
Uses a location contained by an input feature as its output point location.
What I would like to know is how is this point calculated by ArcGIS Desktop and what is its "theoretical" meaning, if that makes sense.