a.rad is used as the proper name Arad, associated with a king and with a location south of it.

The term is consistently a proper name: it identifies Arad in references to its king and to territory south of it.

1. Proper name in royal and geographic references · dominant

   All four occurrences are tagged Noun · Proper. In Numbers 21:1 and Numbers 33:40, the APB renders clauses concerning “the king of Arad” in the Negeb; Joshua 12:14 lists “the king of Arad,” while Judges 1:16 places the wilderness of Judah south of Arad.

   Joshua 12:14Judges 1:16Numbers 21:1Numbers 33:40

In the Anselm Project Bible: The APB presents the name in the phrases “the king of Arad” and “south of Arad.” These phrases show the verse contexts rather than a separate English definition.