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I have a shapefile that contains several polygons which in fact represent the limits of several counties. My task is to find each length of each shared border. So in other words I want to know how many km of border share the A county with the B county and so on. Of course I am looking for a tool to do this automatically. Measuring them manually would take ages. I envision some sort of matrix, but not sure how to do that in ArcGIS.

I use ArcGIS Desktop 10.0 (ArcEditor) .

10x

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Use Intersect tool by setting output_type to "Line". The result will be edges of all input (polygon) layer. Just calculate the length of the output(line) and that is it.

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