frolic vs run around

frolic is a noun but run around is not a noun.

Word	Noun	Adjective	Verb	Adverb
frolic	Yes	No	Yes	No
run around	No	No	Yes	No

As verbs, frolic and run around are synonyms defined as:

Other synonyms of frolic include cavort, disport, frisk, gambol, lark, lark about, rollick, romp, skylark, sport.

frolic (noun)	run around (noun)
gay or light-hearted recreational activity for diversion or amusement

frolic (verb)	run around (verb)
play boisterously	play boisterously