arena vs playing field

arena and playing field both are nouns.

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

As nouns, playing field is a hypernym of arena; that is, playing field is a word with a broader meaning than arena:

Other hypernyms of arena include athletic field, field, playing area.

arena (noun)	playing field (noun)
a playing field where sports events take place	a piece of land prepared for playing a game
a large structure for open-air sports or entertainments	the circumstances under which competition occurs
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