The father of Konstantin, Jakob (Jaagup) Päts (1842-1909), was a housebuilder from Heimtali, near Viljandi.
Konstantin had an older brother Nikolai (1871-1940), three younger brothers Paul (1876-1881), Voldemar (1878-1958) and Peeter (1880-1942) and a younger sister Marianne (1888-1947).