Eruvin 76B

הני מילי בעיגולא, אבל בריבועא בעינן טפי.

This measurement applies only to a circle and the ratio between its circumference and diameter, but with regard to a square that must fit entirely within that circle, we require a circle with a larger circumference. In order for a square of 4 by 4 handbreadths to be entirely contained within a circle, the circumference of the circle must measure more than 12 handbreadths

מכדי, כמה מרובע יתר על העגול ⁠— רביע, בשיתסר סגיא!

The Talmud asks: Now, how much larger is a square than a circle? It is larger by 1/4th. If so, a circle with a circumference of 16 handbreadths at most should suffice.

הני מילי עיגולא דנפיק מגו ריבועא. אבל ריבועא דנפיק מגו עיגולא, בעינן טפי. מאי טעמא? משום מורשא דקרנתא.

The Talmud answers: This statement that a square is larger than a circle by 1/4th applies only to a circle circumscribed by a square, but with regard to a square circumscribed by a circle, we require more, and the difference between the square and the circle is greater. What is the reason for this? It is due to the projection of the corners of the square, as the distance from the center of the square to its corners is greater than the distance from the center to its sides.

מכדי, כל אמתא בריבוע ⁠— אמתא ותרי חומשי באלכסונא, בשיבסר נכי חומשא סגיא.

The Talmud further objects: Since every cubit in the side of a square is a cubit and 2/5ths in the diagonal, a square of 4 by 4 handbreadths has a diagonal of 5 and 3/5ths handbreadths. And since the diameter of a circle equals the diagonal of the square that it encompasses, the circle circumscribing a square of 4 by 4 handbreadths has a diameter of 5 and 3/5ths handbreadths. If that measure is multiplied by 3 to arrive at the circumference of that circle, the result is that a circle with a circumference of 17 handbreadths minus a fifth is sufficient to circumscribe a square of 4 by 4 handbreadths. Why, then, does R' Yoḥanan say that a circular window must have a circumference of 24 handbreadths?