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I am using ArcMap 10.2 and I have a polygon feature that has four sides. On the south side, I would like a buffer of 500 feet and west side, I would like a buffer of 300 ft. I do not need a buffer of the remaining sides (I realize I can just delete those vertices later).

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You can't do this with a polygon in one step. You could create two different buffers, one at 300' and one at 500', then merge and edit them manually.

Another approach would be to convert your polygon layer to a line layer, split the lines at the polygon's 4 corners, then buffer the west side by 300' and the south side by 500'. You'd still need to merge them and manually edit the intersection of the two buffers.

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