radiate v.

1590s, "cross-shaft, straight rod or bar," from Latin radius "staff, stake, rod; spoke of a wheel; ray of light, beam of light; radius of a circle," a word of unknown origin. Perhaps related to radix "root," but de Vaan finds that "unlikely." The classical plural is radii. 

The geometric sense of "straight line drawn from the center of a circle to the circumference" is recorded from 1650s. Meaning "circular area of defined distance around some place" is attested from 1853. As the name of the shorter of the two bones of the forearm from 1610s in English (the Latin word had been used thus by the Romans).

c. 1600, "to cast beams of light upon," from Latin irradiatus, past participle of irradiare "shine forth, beam upon, illumine," from assimilated form of in- "into, in" (from PIE root *en "in") + radiare "to shine" (see radiate (v.)). Meaning "expose to radiation other than light" (originally X-rays) is from 1901. Related: Irradiated; irradiating.