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When using ArcMap (10.1), say I perform an operation like a Polygon Volume calculation.

Are the results in map units or display units (i.e., according to the Data Frame Properties > General tab)?

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It depends which tool you're using, but in my experience (I haven't used every single ArcGIS tool), the output units are usually the same as the units used in whatever the source layer is for your calculation. So for your example, if your polygons are in a coordinate system that uses meters as the linear unit, the polygon volumes will be in cubic meters. The map units and display units are irrelevant in that case.

Some tools allow you to specify the output units, for example if you right-click a numeric field in a layer's attribute table and click Calculate Geometry, you can choose from different units of measure. The Buffer geoprocessing tool is another that lets you choose the units used, although for that one, you're choosing the input units, not the output.

Also, you may be using two different layers to determine volume: a polygon layer to provide the 2D shape and an elevation raster to provide the height values. Those layers may be in different units. In that case, the tool should provide a Z_Factor input which you can use to convert the values in your elevation raster to the same units as those used in your polygon layer. The 3D Analyst Surface Volume tool is one tool that uses a Z_Factor.

If the tool doesn't offer you a choice, the output units are usually the same as the units of the layer you're running the tool on. But it never hurts to check the help for the tool you're using just to make sure.

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