academy vs Gymnasium

academy and Gymnasium both are nouns.

Word	Noun	Adjective	Verb	Adverb
academy	Yes	No	No	No
Gymnasium	Yes	No	No	No

As nouns, Gymnasium is a hypernym of academy; that is, Gymnasium is a word with a broader meaning than academy:

academy: a secondary school (usually private)
Gymnasium: a school for students intermediate between elementary school and college; usually grades 9 to 12

Other hypernyms of academy include lycee, lyceum, middle school, secondary school.

academy (noun)	Gymnasium (noun)
a learned establishment for the advancement of knowledge	a school for students intermediate between elementary school and college; usually grades 9 to 12
a secondary school (usually private)
a school for special training
an institution for the advancement of art or science or literature