dance vs performing arts

dance and performing arts both are nouns.

Word	Noun	Adjective	Verb	Adverb
dance	Yes	No	Yes	No
performing arts	Yes	No	No	No

As nouns, performing arts is a hypernym of dance; that is, performing arts is a word with a broader meaning than dance:

Other hypernyms of dance include diversion, recreation.

dance (noun)	performing arts (noun)
taking a series of rhythmical steps (and movements) in time to music	arts or skills that require public performance
an artistic form of nonverbal communication
a party for social dancing
a party of people assembled for dancing

dance (verb)	performing arts (verb)
move in a pattern; usually to musical accompaniment; do or perform a dance
move in a graceful and rhythmical way
skip, leap, or move up and down or sideways