They’re from the Greek άρχ, via the Latin arx. These words originally meant citadel or fortress, but because such defenses are usually built on heights, came to also mean high or chief.
This is a mess. There is no “Greek άρχ”; there is a Greek prefix άρχ(ι)- (from which our prefix arch(i)- derives), which derives from the verb ἄρχω ‘to lead, rule,’ from Proto-Indo-European *h₂ergʰ-. It has nothing to do with Latin arx, which is from Proto-Indo-European *h₂erk-.