A formal dance was an event during which formally attired attendees danced together.

In 2266, self-proclaimed "Captain" Kevin Riley, while suffering from polywater intoxication and having sealed himself in engineering aboard the USS Enterprise, announced to the vessel's crew that there would be a formal dance in the starship's bowling alley at 1900 hours. When Captain James T. Kirk and an engineering team finally managed to break into engineering, Riley innocently, regretfully told them, "No dance tonight." (TOS: "The Naked Time")

In an ultimately omitted line of dialogue from the final draft script of "The Naked Time", Riley joked that all personnel who attended the formal dance would "have a ball," referencing not only that formal dances were otherwise known as balls, but also that objects called balls were commonly used at bowling alleys, and that "a ball" is an informal term essentially meaning "a good time."
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