The noun G. raith appeared in the Gnomish Lexicon of the 1910s, with glosses “extent, reach; region, sphere, district; scope” (GL/65). It was likely based on the early root ᴱ√RAHA “stretch forward” (QL/32, 78). A similar form ᴱN. rhaith appeared in Early Noldorin Word-lists of the 1920s, but without any gloss (PE13/152).

Neo-Sindarin: For purposes of Neo-Sindarin, I would update this noun to ᴺS. raeth, basing it on the later root ᴹ√RAK “stretch out, reach” with the Sindarin phonological result of aeth < aith from the vocalization of ancient akt(h). I would use it only for the senses “extent, reach, scope”, since there are plenty of later attested words for things like “region, sphere, district”.