宣 xuān to proclaim

to proclaim; to declare; to announce

How it came to be

A roof over a spiral, a hall with a coiled pattern in the floor or ceiling. It was one particular palace building, where announcements were made.

宣布 (xuānbù), to announce, is used for a formal, one-time declaration: 宣布结果 (xuānbù jiéguǒ), announce the results.
宣传 (xuānchuán), to publicize/promote, is an ongoing activity, and is neutral in Chinese (used for marketing and campaigns), unlike the negative connotation "propaganda" often carries in English.
宣誓 (xuānshì), to swear an oath, is used for ceremonial pledges, e.g. taking office.
Formal, official register throughout; not used in casual speech.