late 14c., "measured ground for a combat, joust, race., etc.," in a Scottish source, and according to OED "Until the latter part of the 19thy cent. only in Sc. use;" probably from Old French renc, reng "row, line," from Frankish or another Germanic source, from Proto-Germanic *hringaz "something curved, circle" (from PIE root *sker- (2) "to turn, bend"). But probably much confused in meaning with ring (n.1), also used for "area marked out for a sporting contest."
By 1787 (Burns) as "a sheet of ice measured off for curling;" extended to smooth wooden floors for roller-skating by 1875, to ice surfaces measured for ice hockey by 1896. By 1895 as "building containing a skating rink."