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For my project, I want to calculate the geometry of individual polygons in my layer to find area and perimeter, however it is greyed out when I try to use this tool.

Beforehand, I set the data frame for map and display units to be in meters. I also did not put a projected coordinate system. However, I was able to find which polygons are within a certain distance (ex. 1000 m) from roads using the 'Select by Location' tool.

Why is it that ArcGIS can calculate distance but won't allow me to calculate the geometry of my individual polygons. Am I missing a step here?

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Select by location uses the dataframe environment which has been set in the dataframe properties. Unknown coordinate systems for feature class are the assumed to be identical to the current dataframe coordinate system.

calculate geometry is looking for the feature class properties, therefore it is not allowed if the feature class doesn't have a defined coordinate system. Even using the dataframe coordinate system won't work because there is no conversion from "unknown" to any coordiante system.

It is not recommended to work without a coordinate system, but if you want a workaround you can use the field calculator (with Python parser) which works directly on the vertices' coordinates.

!Shape.Area! 

and

!shape.length!

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