A ROUND SUKAH
(R. Yochanan): A round Sukah is Kosher if its circumference is large enough to contain 24 people.
Question: R. Yochanan's requirement aligns with no other opinion, even the largest opinion (Rebbi, who says that any Sukah which is not 4x4 Amos is Pasul)!?
A person can sit in a space of 1 Amah (thus R. Yochanan's 24 people must be 24 Amos).
But we know that the relationship of the circumference to the diameter is 3 to 1.
Then taking the largest requirement for the width of a Sukah, Rebbi's (four Amos), it should suffice to have a circumference of only 12 Amos (not 24)!
Answer: This relationship between the circumference and diameter applies only to a circle.
The perimeter of a square, though, is larger than the ratio of 3:1.
Thus in order to have a circle which has a circumference equal to the perimeter of a 4 X 4 Amos square, the circle must be larger than 12 Amos in circumference].
Question: But that still does not give us 24!?
A square is a fourth (25%) larger than a circle (Rashi: a circular Amah is surrounded by a 3 Amah circumference, while a square Amah is surrounded by a 4 Amah perimeter).
It should then suffice to have a circumference of only 16 Amos (and still not 24)!
Answer: This formula (that a square is 25% larger than a circle) applies only to a circle which is inscribed within a square.
The diameter of the inscribed circle is equal to the length of a side of the square.
However, the size of a circle which is circumscribed around a square (i.e. the square is inside the circle, such that the diameter of the circle is equal to the diagonal of the square) must be larger, because of the arcs of the circle.
Question: But that would still give us only 16 4/5 Amos!?
The relationship of a side of a square to its diagonal is 1 to 1 2/5.
Thus, the diagonal of the square, which is the same as the diameter of the circle around it, is 4 X 1 2/5 = 5 3/5.
Since the circumference of a circle is 3 times its diameter, the circumference of this circle is 5 3/5 X 3 = 16 4/5).
It should then suffice to have a circumference of only 16 4/5 Amos (and still not 24)!
Answer: R. Yochanan was not exact in his number.
Question: We only say that one was not exact in his number when the difference is small, not when it is so large (24 vs. 16 4/5)!?
Answer (Mar Kashisha): R. Yochanan's number was based on the assumption that 3 persons fit in 2 Amos (not one person in one Amah).
Question: If so, R. Yochanan was saying that the size of the circumference of the round Sukah needs to be only 16 Amos (3 person in 2 Amos = 24 persons in 16 Amos), yet we concluded that the circle around a square which is 4 X 4 Amos must have a circumference of 16 4/5 Amos!
Answer: R. Yochanan was not exact in his number.
Question: We only say that one was not exact in his number when he was rounding off to a figure that would be stringent.
Here R. Yochanan (who says that the circumference must be 16) is rounding off to a figure that would be lenient.
This would allow for a smaller Sukah than actually required!
Answer (R. Asi): R. Yochanan was not including the place where the person sits, which makes the measured circumference 18, not 24.
R. Yochanan's number was indeed based on the assumption that one person fits in one Amah.
However, R. Yochanan was not including in the size of the Sukah the place which the sitting person occupies.
Rather, the Sukah is measured from within (or inside) the place where the person sits.
Hence, if 24 people sit around the Sukah and we do not include the Amah in which they sit, the circumference of the circle comes out to be 18 (Rashi: we subtract two Amos from the diameter of the circle [one Amah on each 'side' of the Sukah] leaving a diameter of 6. The circumference of such a circle is 18).
Question: But we determined that it is enough to have a circumference of 16 4/5 Amos (and not 18)!
Answer: That is where R. Yochanan was not exact in his number, and his inexactness was a Chumra.
THE GEOMETRICAL FORMULAE OF THE RABANAN D'KESARI
The circle that comes out from within a square is a fourth (25%) smaller than the square around it.
The square that comes out from within a circle is one half (50%) smaller than the circle around it.
Question: This is not so, for we see that the circle circumscribed around a square is not so much larger. (Question remains unanswered. See Insights)
FULFILLING THE MITZVAH WITH SUKOS BUILT FOR OTHER PURPOSES
(R. Levi citing R. Meir): The inner Sukah of a Potter (used for residence) is not Kosher as a Sukah, and it requires a Mezuzah, because it is the place in which he lives all year long.
The outer Sukah (used as his shop and store) is a Kosher Sukah, and is exempt from a Mezuzah, because it is not normally used as his residence.
Question: Why is the outer Sukah exempt from a Mezuzah (it should be like an entrance hall to a house, which requires a Mezuzah)!?
Answer: Since the whole structure is permanent, we only go as far as to obligate the residence Sukah in Mezuzah, not to consider the outer Sukah as its entrance hall.
(Beraisa): The Sukos of GaNVa"Ch - The Sukos of Goyim, Nashim, Behemah, Kusim,- and any Sukah (not made for the Mitzvah) are Kosher, as long as it has Halachic Sechach.
Question: What is particular about these as needing Halachic Sechach?
Answer (R. Chisda): The Sechach must have been placed on the Sukah to provide shade (not only for privacy).
Question: What is added by "any Sukah?"
Answer: It includes the Sukos of RaKVa"Sh, as in the following Beraisa:
(Beraisa): The Sukos of RaKVa"Sh - The Sukos of Ro'im, Kayatzim, Burganin, Shomrei Peros,- and any Sukah are Kosher, as long as it has Halachic Sechach.
Question: What is Halachic Sechach?
Answer (R. Chisda): The Sechach must have been placed on the Sukah for shade (not only for privacy).
Question: What is "any Sukah" adding?
Answer: It adds the Sukos of GaNVa"Ch as we learned in the earlier Beraisa.
Question: Why did each Tana specify one group of Sukos and add the other group with "any Sukah?"
Answer: Each specifies the group of Sukos more likely to be considered Kosher, and adds the less likely group in the "any Sukah."
The first Tana held that the Sukos of GaNVa"Ch have the Halachic advantage of being more permanent, compared to the RaKVa"Sh Sukos which are, at times, not even left standing for seven days, (so he needed to add the RaKVa"Sh Sukos as Kosher).
The second Tana held that the Sukos of RaKVa"Sh have the advantage of being used by people who are Chayav in the Mitzvah of Sukah (and he thus adds the Sukos of GaNVa"Ch which are used by those who are Peturim).