Corner areas and rooms in a Sukkah [see illustration]:[1]
A Sukkah that contains an extension area [i.e. “corner” in Halacha] that is less than 7×7 Tefachim, then that extension is invalid, and does not join the rest of the Sukkah. [If either the length or width of the extension area is less than 7 Tefachim, it is invalid.[2] However, one may be lenient to have one’s table situated in the area of this extension, even if it is less than 7×7.[3] Likewise, rooms that are separated with Mechitzos are invalid if they contain a dimension that is less than 7×7 Tefachim.[4] Thus, not only must the entire Sukkah be 7×7 Tefachim, but any area of the Sukkah that is separate from the other areas must also contain a dimension of 7×7 Tefachim.]
[1] M”A 634:1; M”B 634:1; Biur Halacha 634:1 “Sukkah” in length; See Bigdei Yesha 634; Bikureiy Yaakov 634
Other opinions: Some Poskim negate the ruling of the M”A and rule that an extension of the Sukkah is valid just like the Sukkah itself and does not need to have its own 7×7 dimension. [Chazon Ish 144:6; Orchos Rabbeinu 2:29; See Bigdei Yesha 634; Bikureiy Yaakov 634]
The status of a Tzuras Hapesach in middle of Sukkah: If a Tzuras Hapesach is made in middle of a Sukkah, it is disputed as to whether it splits the Sukkah in two, and thus requires that both sides contain 7Ă—7 and 3 Kosher walls for it to be Kosher. [See Minchas Yiotzchak 10:50 who rules its valid; Piskeiy Teshuvos 630:4 in name of Poskim that perhaps it is invalid]
[2] Biur Halacha 634:1 “Sukkah”, in negation of Derech HaChaim
Other opinions: Some Poskim rule that a if the length of the extension is not 7 Tefachim it is valid, so long as the width is 7 Tefachim, being that one can join the rest of the Sukkah to its length. [Derech HaChaim, brought in Biur Halacha ibid; Chazon Ish 144:6; Chelkas Yaakov 3:16; See Piskeiy Teshuvos 634:1]
[3] Biur Halacha 634:1 “Sukkah” in name of Bikureiy Yaakov
[4] See Q&A!