As it turns out, I need to revisit my centroid plugin for SketchUp. While it accurately calculates area properties for a face that does not contain internal (and fully enclosed) openings, it apparently gets it wrong as soon as an enclosed opening is present (like the face in the top-right corner of the image above).
To diagnose the issue, I made a little script to number vertices for a face. It adds the vertex numbers as well as the face normal vector to a group and displays them. Just in case anyone else finds this useful, here is the download. Install the file like any other SketchUp plugin (i.e. drop it into the “Plugins” folder in the SketchUp installation).
The script adds a menu option under the “Tools” menu. You can also change in the source which vertices should be displayed (all or “outer loop” only).