הָנֵי מִילֵּי בְּעִיגּוּלָא אֲבָל בְּרִיבּוּעָא בָּעֲיָא טְפֵי The Gemara answers: This applies only in the middle of the circle that has a circumference of twelve cubits, as the diameter of the circle is four cubits; but in order for a square inscribed within a circle to have a perimeter of sixteen cubits, the circle requires a circumference that is more than twelve cubits.
מִכְּדֵי כַּמָּה מְרוּבָּע יוֹתֵר עַל הָעִיגּוּל רְבִיעַ בְּשִׁיתְּסַר סַגִּי The Gemara asks further: Now, by how much is the perimeter of a square inscribing a circle greater than the circumference of that circle? It is greater by one quarter of the perimeter of the square. If that is the case, a circle with a circumference of sixteen cubits is sufficient. Why, then, does Rabbi Yoḥanan require a circumference of twenty-four cubits?
הָנֵי מִילֵּי בְּעִיגּוּל דְּנָפֵיק מִגּוֹ רִיבּוּעָא אֲבָל רִיבּוּעָא דְּנָפֵיק מִגּוֹ עִגּוּלָא בָּעֵינַן טְפֵי מִשּׁוּם מוּרְשָׁא דְקַרְנָתָא The Gemara answers: This statement with regard to the ratio of the perimeter of a square to the circumference of a circle applies to a circle inscribed in a square, but in the case of a square circumscribed by a circle, the circle requires a greater circumference due to the projection of the corners of the square. In order to ensure that a square whose sides are four cubits each fits neatly into a circle, the circumference of the circle must be greater than sixteen cubits.
מִכְּדֵי כׇּל אַמְּתָא בְּרִיבּוּעָא אַמְּתָא וּתְרֵי חוּמְשֵׁי בַּאֲלַכְסוֹנָא בְּשִׁיבְסַר נְכֵי חוּמְשָׁא סַגִּיא לָא דָּק The Gemara calculates precisely how much greater the circumference must be in order to circumscribe the four-by-four-cubit square. Now, in every square whose sides each measure one cubit its diagonal measures one and two-fifths cubits, and in a circle that circumscribes a square, the diagonal of the square is the diameter of the circle. In this case, the circumscribed square measures four by four cubits; therefore, the diagonal of the square, which is the diameter of the circle, measures five and three-fifths cubits. Since the Gemara calculates the circumference of the circle as three times its diameter, a circular sukka with a circumference of seventeen cubits minus one-fifth of a cubit should be sufficient. The Gemara answers: Rabbi Yoḥanan was not precise and rounded the dimensions of the circular sukka to a number higher than the absolute minimum.
אֵימוֹר דְּאָמְרִינַן לָא דָּק פּוּרְתָּא טוּבָא מִי אָמְרִינַן לָא דָּק The Gemara wonders: Say that we say that the Sage was not precise when the difference between the number cited and the precise number is slight; however, when the difference is great, do we say the Sage was not precise? After all, Rabbi Yoḥanan stated that the minimum measure is twenty-four cubits, a difference of more than seven cubits.
אֲמַר לֵיהּ מָר קַשִּׁישָׁא בְּרֵיהּ דְּרַב חִסְדָּא לְרַב אָשֵׁי מִי סָבְרַתְּ גַּבְרָא בְּאַמְּתָא יָתֵיב תְּלָתָא גַּבְרֵי בְּתַרְתֵּי אַמְּתָא יָתְבִי כַּמָּה הָווּ לְהוּ שִׁיתְּסַר אֲנַן שִׁיבְסַר נְכֵי חוּמְשָׁא בָּעֵינַן לָא דָּק Mar Keshisha, i.e., the elder, son of Rav Ḥisda, said to Rav Ashi: Do you hold that when a man sits, he sits and occupies one cubit, and consequently a sukka that seats twenty-four people must have a circumference of twenty-four cubits? In fact, three people sit and occupy two cubits. The Gemara asks: How many cubits are there in the sukka required by Rabbi Yoḥanan? There are sixteen cubits. But we require a sukka with a circumference of seventeen cubits minus one-fifth, as calculated above. The Gemara answers: He was not precise and rounded the figure down to the lower whole number; actually, the required circumference is four-fifths of a cubit larger.
אֵימוֹר דְּאָמְרִינַן לָא דָּק לְחוּמְרָא לְקוּלָּא מִי אָמְרִינַן לָא דָּק The Gemara rejects this explanation: Say that we say that the Sage was not precise when the result is a stringency, e.g., he required a sukka whose dimensions are greater than the minimum required dimensions; however, when the result is a leniency, do we say the Sage was not precise? In that case, the lack of precision will lead to establishing a sukka whose dimensions are smaller than the minimum requirement.
אֲמַר לֵיהּ רַב אַסִּי לְרַב אָשֵׁי לְעוֹלָם גַּבְרָא בְּאַמְּתָא יָתֵיב וְרַבִּי יוֹחָנָן מְקוֹם גַּבְרֵי לָא קָחָשֵׁיב Rav Asi said to Rav Ashi: Actually, a man sits and occupies one cubit, and Rabbi Yoḥanan is not factoring the space that the men occupy in his calculation. In other words, to this point, the assumption has been that Rabbi Yoḥanan calculated the circumference of the sukka required to seat twenty-four people. Actually, he merely calculated the circumference of the inner circle formed by the twenty-four people seated.
כַּמָּה הָווּ לְהוּ תַּמְנֵי סְרֵי בְּשִׁיבְסַר נְכֵי חוּמְשָׁא סַגִּיא הַיְינוּ דְּלָא דָּק וּלְחוּמְרָא לָא דָּק The Gemara asks: How many cubits are there in the circumference of the inner circle formed by a circle of twenty-four people? There are eighteen cubits. Based on the principle that for every three cubits of circumference there is one cubit of diameter, the diameter of a circle whose outer circumference surrounds twenty-four people is eight cubits. To calculate the circumference of the inner circle, subtract from the diameter the space occupied by two people, each sitting at one end of the diameter. The result is a diameter of six cubits. Based on the above principle, a circle with a diameter of six cubits will have a circumference of eighteen cubits. However, a circumference of seventeen cubits minus one-fifth of a cubit should be sufficient. The Gemara answers: This is the case where he was not precise, and in this case he is not precise when the result is a stringency, as instead of sixteen and four-fifths, Rabbi Yoḥanan required eighteen cubits.
רַבָּנַן דְּקֵיסָרִי וְאָמְרִי לַהּ דַּיָּינֵי דְקֵיסָרִי אָמְרִי עִיגּוּלָא דְּנָפֵיק מִגּוֹ רִיבּוּעָא רִבְעָא The Sages of Caesarea, and some say that it was the judges of Caesarea, said that Rabbi Yoḥanan’s statement could be explained using a different calculation: The circumference of a circle inscribed in a square is one-quarter less than the perimeter of the square,