[Avodah] Dimensions of a circular sukkah

[Kenneth Miller]

If I have a sukkah which is the minimal 7 tefachim square, the diagonal will be about 9.8995 tefachim long. (You can reach that figure either by multiplying 7 by the square root of two, or by using the Pythagorean Theorem to take the square root of 98.) If I use that diagonal as the diameter of a circle, that circle will be the smallest possible circle to enclose that square, and its circumference (calculated by multiplying the above figure by pi) will be about 31.1 tefachim.

The above calculation shows why I was very surprised to find Mishna Berurah 634:4 (citing Beis Yosef) giving the circumference of that circle as only 29.4 tefachim. In the Shaar Hatziun there, he explains the calculation: Seven is multiplied by 1.4 to get the length of the diagonal. That is the diameter of the circle, and is multiplied by 3 to get the circumference. This calculation gives 29.4.

This surprises me on several levels. First, although I do recall Chazal using these values for sqrt (2) and for pi, my recollection is that they did NOT take this values as exact, but usually appended the words "and a little more". Unfortunately, I have no sources to offer; can any of the chevra support or refute my claim?

Second, was the Chofetz Chaim's mathematics so rudimentary as to accept 3 as pi for a d'Oraisa? He was rather recent, as acharonim go. Am I expecting too much of him?

Third, this seems like such an easy thing to check experimentally. You don't need exact tefachim;  just make a seven-unit square-circle of any size, and measure it. I think this point is especially strong if you look at the perimeter of that square: The figure of 29.4 is closer to the square's perimeter of 28 than to the circle's circumference of 31.1. Any figure that can be surrounded by a 29.4 string is either a grossly misshapen circle, or it is a circle that can surround a square whose sides are about 6.62 long. That's more than a third of a tefach short!

I *am* familiar with the concept that we've often discussed here, that a working legal system needs to incorporate estimates and calculations that are feasible to the average person, and must sometimes choose expedient over accurate. Still, this case seems to be rather extreme. Is 3 really an accepted value for pi in halacha l'maaseh, as opposed to the much closer 3 1/6? Can anyone comment?

Akiva Miller
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