arena theater vs theatre

arena theater and theatre both are nouns.

Word	Noun	Adjective	Verb	Adverb
arena theater	Yes	No	No	No
theatre	Yes	No	No	No

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

arena theater: a theater arranged with seats around at least three sides of the stage
theatre: a building where theatrical performances or motion-picture shows can be presented

Other hypernyms of arena theater include house, theater.

arena theater (noun)	theatre (noun)
a theater arranged with seats around at least three sides of the stage	a building where theatrical performances or motion-picture shows can be presented
	the art of writing and producing plays
	a region in which active military operations are in progress