'arc-length' is a compound formed from 'arc' and 'length'. 'arc' ultimately originates from Latin, specifically the word 'arcus', where 'arc-' meant 'bow' or 'arch'; 'length' originates from Old English 'lengðu', where 'lengð' related to 'long' or 'length'.

'arc' came into English via Old French 'arc' (from Latin 'arcus'), while 'length' developed from Old English 'lengðu' into Middle English 'lengthe' and eventually modern English 'length'; the compound 'arc length' arose in mathematical usage to denote the length along an arc.

Individually, 'arc' originally referred to a bow or curved shape and 'length' to an extent measured in one dimension; combined in mathematical contexts they came to specifically mean 'the distance measured along a curve' (the modern technical meaning).