Sukkah 8A

הני מילי בעיגולא, אבל בריבועא ⁠— בעיא טפי.

The Talmud answers: This applies only in the middle of the circle that has a circumference of 12 cubits, as the diameter of the circle is 4 cubits; but in order for a square inscribed within a circle to have a perimeter of 16 cubits, the circle requires a circumference that is more than 12 cubits.

מכדי כמה מרובע יותר על העיגול ⁠— רביע, בשיתסר סגי?

The Talmud 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 1/4th of the perimeter of the square. If that is the case, a circle with a circumference of 16 cubits is sufficient. Why, then, does R' Yoḥanan require a circumference of 24 cubits?

הני מילי בעיגול דנפיק מגו ריבועא, אבל ריבועא דנפיק מגו עגולא ⁠— בעינן טפי, משום מורשא דקרנתא.

The Talmud 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 4 cubits each fits neatly into a circle, the circumference of the circle must be greater than 16 cubits.

מכדי כל אמתא בריבועא אמתא ותרי חומשי באלכסונא, בשיבסר נכי חומשא סגיא? לא דק.

The Talmud calculates precisely how much greater the circumference must be in order to circumscribe the 4-by-4-cubit square. Now, in every square whose sides each measure 1 cubit its diagonal measures one and 2/5ths 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 4 by 4 cubits; therefore, the diagonal of the square, which is the diameter of the circle, measures 5 and 3/5ths cubits. Since the Talmud calculates the circumference of the circle as 3 times its diameter, a circular sukka with a circumference of 17 cubits minus 1/5th of a cubit should be sufficient. The Talmud answers: R' Yoḥanan was not precise and rounded the dimensions of the circular sukka to a number higher than the absolute minimum.