The [online help][1] indicates that you must check the box to be INSIDE so that a point inside the polygon will be used for the output. As @DanC commented: > The center of gravity can in fact fall outside the polygon. Think of a > donut shape, with an empty hole in the middle of the circle polygon. > The center of gravity would be there even though the polygon doesn't > extend there. [1]: http://webhelp.esri.com/arcgisdesktop/9.3/index.cfm?TopicName=Feature_To_Point_%28Data_Management%29