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The rim of a coin can be found on either side. In fact, a coin can be said to have two rims, one on the obverse and one on the reverse. The rim is the up-raised part of the coin that completely encircles the diameter on both front and back. On U.S. coins, the rim is usually very thin, but on Presidential Dollars it is much wider, the purpose being to allow blind people to tell the dollar apart from the quarter dollar coin by touch alone. (This standard was created back when the Susan B. Anthony dollar also had the reeded edge, so the lack of reeding wasn't enough for the blind to make the distinction.)
Do not confuse Rim with an Edge of a coin