Conrad We use the same in-house cadastral feed projected into web mercator auxillary sphere to populate our intranet map and for desktop background to aid in location and estimation. We understand that measures will be a little off, maybe up to 10%. However, it is 200% wrong and quite misleading in ArcMap desktop, which is not the case in our intranet ArcGIS for Server map. A township measures approximately 36 sq.mi in the web-based browser map but in desktop that same township measures approximately 72 sq.mi. No one's buiding a fence but still the error is astounding!! Let's get it fixed. Below is part of the response I received from ESRI.
From ESRI Tech support
As we discussed I have logged BUG-000088279 [Geodesic Areas cannot be calculated in the Measure Tool in ArcMap, if the Data Frame is set to a Projected Coordinate System. This is a significant problem particularly using Web Mercator.]
I do have another suggestion regarding this issue also. If you post this on the Ideas forum, where other users can vote on the issue, it will raise the profile of the problem. The Ideas forum link is: