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I have two layers: One of the Misissippi River Basin and one of the USA counties within the 48 states.

I'm having no trouble selecting the counties within the Basin, but I also need to select the portions of the counties only partially within the basin. I know how to select the entirety of the counties that are partially within, but I want to select only the portions that are within the basin shapefile so I can calculate the percentage of those counties that are within the MRB.

Edit for more info: Take an examplehere that shows a selected county - the counties layer is red and part of the county selected contains a portion of the Mississippi river basin shapefile (green). Rather than select the whole county when I select by location (I've selected those that intersect the MRB shapefile, those that are wholly within it, and those only partially within it - I still get entire counties) I would like to select only the portion of each county that contains the MRB shapefile shown in green.

Furthermore, I would like to retain the borders of each county - My end goal is to extract county-level data from a raster, but only the portions of counties within the River Basin.

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    Welcome to the GIS SE. Could you post a picture / snapshot of what the data looks like? It sounds like you need to use the "Select by Location Tool" and the Spatial selection method set to "touch the boundary of the source feature layer" – Keagan Allan Oct 21 '18 at 23:28
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    You can use select layer by location with the option intersect to selct all counties that touch the river basin polygon. As for calculating percentage, assuming you're using shapefiles, add a field called orig_area, calculate geometry as the shape area in a suitable projected coordinate system then use the intersect tool, add a field called Int_area and Pcnt_In, calculate geometry on int_area in the same CRS and field calculate Pcnt_in = Int_area / orig_area (decimal percent, multiply by 100 to express as a percentage value). Does that help? Are you doing this interactively or in python? – Michael Stimson Oct 21 '18 at 23:57
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You can clip the counties by Basin and then join the table with the original one to calculate the portion of the area.

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