According to the ArcGIS Resource Center:
Both input features and near features can be point, multipoint, line,
So you should be able to use the Near Tool for both polygon layers as Rayner suggested but without needing to convert them. The Near tool can be found in the Proximity toolset of the Analysis tools in ArcToolbox.
Just set the Counties layer as the input feature and the Georgia layer as the near feature. The following attributes would then be added to the Counties layer:
- NEAR_FID—Stores the feature ID of the nearest feature.
- NEAR_DIST—Stores the distance from the input feature to the nearest feature. The value of this field is in the linear unit of the input's
I take it the NEAR_DIST values are what you need?
Extracting the Georgia boundaries from the states shape file
As for your data, you can extract the Georgia boundary from the states shapefile. Go to your states shapefile's attribute table and select Georgia.
The Georgia state should be highlighted. You can then extract it by right-clicking the states layer name in the table of contents and clicking Data > Export Data. This displays the Export Data dialog box. You can then export just the Georgia boundary by choosing the Selected features option from the dropdown menu.
You can then use the resulting Georgia layer as the Near Feature.