playing area vs arena

playing area and arena both are nouns.

Word	Noun	Adjective	Verb	Adverb
playing area	Yes	No	No	No
arena	Yes	No	No	No

As nouns, arena is a hyponym of playing area; that is, arena is a word with a more specific, narrower meaning than playing area:

playing area (noun)	arena (noun)
a piece of land prepared for playing a game	a playing field where sports events take place
	a large structure for open-air sports or entertainments
	the central area of an ancient Roman amphitheater where contests and spectacles were held; especially an area that was strewn with sand
	a particular environment or walk of life