From Middle English rim, rym, rime, from Old English rima ("rim, edge, border, bank, coast"), from Proto-Germanic *rimô, *rembô, from Proto-Indo-European *rem-, *remə-. Cognate with Saterland Frisian Rim ("plank, wooden cross, trellis"), Old Saxon rimi, Icelandic rimi ("a strip of land").
From Middle English rim, rym, ryme, reme, from Old English rēoma ("membrane, ligament"), from Proto-West Germanic *reumō.