Shemini Atzeret is the day that occurs at the close of Feast of Tabernacles (Lev 23:36) Which occurs  on the 22nd day of the seventh month because when you count from the 15th day of the seventh month, which starts the Feast of Tabernacles, it brings you to the 22nd day of the seventh month, which is the eighth day. We are to have a holy convocation and it should be a Sabbath no servile work shall be done (Lev. 23:36, 39; Num. 29:35).  All the names for Shemini Atzeret in the scriptures are, Solemn Assembly (Lev. 23:36) and The Eighth day (Lev. 23:36,39). Events that took place during the seventh month on Shemini Atzeret are, Solomon and Israel kept Sukkot and the eighth day. (2 Chron. 7:8-9), Ezra read the Law Day by day until the last day of the feast of the tabernacles and they kept the eighth day (Nehemiah 8:18). HaMashiach went back to the temple and sat down and taught the people and the women who committed adultery was brought in before HaMashiach to be brought up on charges (John 8:1-11)

Shemini Atzeret/Eighth day of Assembly