'high-arched' is a Modern English compound of the adjective 'high' and the past participle 'arched' (from the verb 'arch'). 'High' originates from Old English 'heah'. 'Arch' (verb/noun) traces back ultimately to Latin 'arcus' meaning 'bow' or 'arc'.

'high' developed from Old English 'heah' into Middle and then Modern English 'high'. 'Arch' appears in Middle English (from Old English and Old French influences) ultimately from Latin 'arcus'; the adjective form 'arched' (past participle of 'arch') combined with 'high' in Modern English to form the compound 'high-arched'.

Initially the components meant 'high' and 'arched/curved' respectively; the compound has retained the literal sense of 'having a high arch' and has remained semantically stable, used specifically for anatomical or shape descriptions.