Temporarily, the Talmud answers that the previous formula was true only for a circle. However, the perimeter of a square is more than three times its width. The round sukkah must have a circumference that would equal a square that is four cubits wide. So the circumference would indeed have to be more than 12 cubits.
As we shall see in tomorrow’s section, the math is still problematic.
Today’s section continues to deal with the size of the round sukkah. Yesterday we concluded that the perimeter of the square needed to be greater than a circle. But today we see that it still won’t get us from a circumference of 12 to a circumference of 24, the number required by R. Yohanan for the round sukkah to be kosher.
A square is only 25 percent larger than an inscribed circle. Thus if the circle has a diameter of 4, its circumference is 12 (assuming that pi is 3). A square placed around this circle will have sides that are 4 cubits (= to the diameter), leading to a circumference of 16 (4 x 4). So why then does the circle need to be 24 cubits?
The above was assuming that the square was around the circle. But if R. Yohanan was talking about a round sukkah that could fit inside a sukkah of four squared cubits in it, the sukkah must be bigger because of the projection in the corners (the places where the circle doesn’t fill the square.
The problem is that this still doesn’t add up to 24. According to rabbinic calculation, a hypotenuse is 1.4 times the side of a square (in reality it is the square root of 2, but 1.4 is close). So the diameter of the circle (which is equal to the hypotenuse of the inscribed square) is 4 x 1.4—5.6. That would mean that the circumference of the circle is 16.8 (3 x 5.6). We still have not gotten too close to 24.
The Talmud answers that R. Yohanan wasn’t precise in his figure. Admittedly, not a particularly satisfying answer.
More math! We continue to try to figure out how R. Yohanan came up with the number 24 for the required circumference of the round sukkah. Yesterday’s section ended with the conclusion that the sukkah really only needed to have a circumference of 16.8 cubits but that R. Yohanan was simply approximating when he said it needed to be 24 cubits.
If R. Yohanan gave a number that was close to 16.8, we could say that he was issuing an approximation. But 24 cubits is just too far off of 16.8. It’s hard to imagine that he was simply approximating. Thus we are back to square one-why does the circular sukkah have to be so large.
Mar Kashisha now adjusts one of the key figures in the original calculation. We thought that each person takes up 1 cubit, but in reality three men can fit into 2 cubits. So for 24 men to be able to sit around this sukkah, they only need 16 cubits. This is still not exactly what R. Yohanan said, but it’s close. R. Yohanan’s figure is 16 and the size of the sukkah that fits around a 4 cubits square sukkah is 16.8.
The problem is that R. Yohanan’s approximation leads to a leniency. R. Yohanan says it only needs to be 16 cubits around so that 24 people could sit around it. But in reality it should be slightly bigger.
R. Assi now changes the assumption we have been making all along that the people were sitting outside of the sukkah. If each person takes up a cubit’s space, this leads to the diameter being 8 cubits (1/3 of the circumference). But if they sit inside the sukkah, then we can reduce a cubit in each direction, bringing the diameter down to 6 cubits. This leads to a circumference of 18. So if he really needed to require 16.8, the circumference that would allow an inner circle of 16, R. Yohanan was slightly stringent. And it is okay to be offer an approximation if it leads to a stringency.