Clarence Angier was a member of the Birmingham Country Club in Birmingham, Alabama. He was the second oldest child born to Nedom Angier (1814-82) and Elizabeth Herring Angier (1827-1902). His father was mayor of Atlanta from 1877-79, and, as Union sympathizers, the family had to flee Atlanta during the Civil War. Clarence Angier married Maria Louisa Eastman, and they had three children. Angier placed 72nd in qualifying at the 1904 Olympic golf tournament, bettering only two people who completed play.