I have been banging my head for several hours...it appears that there is currently no method/option to automatically calculate the area of a polygon stored in a hosted feature layer in ArcGIS Online without creating a custom application. The measurement tool is cumbersome when dealing with more than one polygon. My current work around is to extract the feature service, calculate in desktop, and re-publish. Someone please tell me I am missing something.
Senior GIS Analyst
Geodata Services, Inc.