Can one rely on the approximate value of Pi for Halacha?

Chazal teach us that if one wants to know the diameter of a circle he needs to divide its circumference by 3. If the circumference of a circle is 3 then its diameter is 1, but is that accurate or an approximation?

עירובין דף יד

כל שיש בהיקפו שלשה טפחים יש בו רחב טפח מנא הני מילי? – אמר רבי יוחנן, אמר קרא : ויעש את הים מוצק עשר באמה משפתו עד שפתו עגל סביב וחמש באמה קומתו וקו שלשים באמה יסב אתו סביב

Chazal tell us that the source for this is from one of the vessels Shlomo Hamelech built for the Bais Hamikdash. The pasuk tells us it was ten amos from one brim to the other and it had a circumference of 30 amos.

One of the first formulas one learns in math is the concept of pi, or π. It is defined as the ratio of a circle’s circumference to its diameter, and it also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. What the exact value of pi is not known but we know that the Egyptian and Babylonian approximations for pi, 3.16 and 3.125 respectively, were established some five centuries before the time of Shlomo HaMelech. 3.14 was used by the famous Greek mathematician Archimedes. Today for all day to day calculations the rounded number 3.14 is used. In reality, this number is not the exact number because pi is infinite.

The obvious question is that even in the ancient times it was known that pi is not exactly 3, so why did Chazal tell us that the circumference to diameter ratio is 3? It is virtually impossible that Chazal were not aware that the result is more than 3. This of course has tremendous halachic ramifications. For as we see in our sugya, the koreh needs to be a tefach wide. If we rely on the 3 ratio and not the 3.14 then if one uses a log which is 3 tefachim around the circumference it would be assumed that according to Chazal it is one tefach wide and therefore considered to be an eligible koreh. In reality the diameter is only .95 tefachim wide since to ratio to be divided is 3.14. This question is mentioned by Tosfos when he remarks that these calculations are not exact according to the mathematicians. 

To answer this we need to ask another question. Chazal tells us the ratio is 3, and they bring proof from the pasuk by Shlomo Hamelech. This seems difficult to understand, why would we need proof from something we can measure ourselves. It is a math question, something which can be analyzed by measuring and calculating. We would not search for proof in the Torah for 1+1=2 as we would understand that it is a calculation and does not require proof. Why then would we need proof for pi?

The Tosfos HaRosh explains that in reality Chazal knew what was commonly known in their times that Pi is not exactly 3. However Chazal believed that when it comes to calculating the size for halacha one can rely on the simple ratio of 3 and would not be required to calculate with the more complicated number of 3.14. The proof for this is from the pool that Shlomo Hamaelech built. The Torah tells us it was 10 amos across and 30 amos around. Although mathematically this is impossible as this would only be possible if pi is 3, still the Torah describes it in a way that it seems the ratio of pi is 3. Hence we have proof that one can rely on the ratio of 3 for pi and not 3.14.

This idea is further elaborated by the Chazon Ish. The Chazon Ish explains that we find a few calculations in Chazal that seem simplified and not exact, such as the idea that every square is a quarter larger than a circle within it. Or the measurements of calculating the hypotenuse that it is two fifths larger than its side. So too, when it comes to figuring out the calendar, we follow the opinion of Shmuel that every year is longer than the year before by a quarter of a day, although in reality none of these calculations are exact. 

The Chazon Ish explains that none of these calculations are exact, but all the shiurim we have were given by Moshe at Sinai. When the shiurim were given, the idea that these calculations can be simplified was included. Since the Torah was given for everyone, even for simple minded people, therefore the Torah does not expect them to need to make difficult calculations to figure out how they can do a mitzvah properly. It is enough if the number is approximately similar to the correct number. The proof for that is from the vessel that Shlomo Hamelech built. 

A slightly different approach is given by the Rambam in Peirush HaMishnayos: “The ratio of the diameter to the circumference of a circle is not known and will never be known precisely. This is not due to a lack on our part but this number [pi] cannot be known because of its nature, and it is not in our ability to ever know it precisely. But it may be approximated to three and one-seventh. So any circle with a diameter of one has a circumference of approximately three and one-seventh. But because this ratio is not precise and is only an approximation, they [the rabbis of the Mishnah and Talmud] used a more general value and said that any circle with a circumference of three has a diameter of one, and they used this value in all their Torah calculations”

It seems that the Rambam is adding an additional reason for why Chazal gave the ratio as 3. Chazal knew that pi was 3.14 and not 3. However, no value of pi that we have is ultimately the exact one as it is an infinite number. At some point we would be obligated to choose one number, and the question would be what that number is. Chazal chose the simple number of 3 rather than 3.14.

As Morris Engelson writes (Hakira vol 22): “What value of pi do we need for ordinary, common daily use? The complete value of pi is not available to us. This is because pi is an irrational number and is impossible to write down its exact value no matter how many decimal places or fractional designation one chooses. Whether we choose π=3, or π=3.14159, or pi to a billion decimal places, all are approximations. π = 3 is the simplest approximation. This introduces an error of just less than 5%. Is that acceptable? It depends on the approximation’s purpose. Today we usually designate π=3.14 for common usage. But would this approximation be useful without the aid of a calculator? Try finding the product of 7 and 3.14 without a calculator and modern writing implements. Compare that approximation with the simple 7×3=21. Is this simplicity worth the 5% error? Many would say it is. Indeed, Chazal appear to agree that 5% is an acceptable error”.

The Tashbatz (Vol 1:165) adds that the Rambam agrees that the reason Chazal knew that they can use a simplified ratio of pi is because we have proof from the pool that Shlomo Hamelch built.

Can one rely on this to use for halacha?
Is one allowed to use pi=3 when the halacha requires something to be a certain size?

The Mishnah Berurah writes (SH”T 372:18) that it seems from the words of the Rambam that when it comes to Mitzvah d’Rabbanan one can definitely rely on pi=3. It is implied that by a d’Oraysa one should use a more exact number where pi is rounded to 3.14. 

I would propose that the question of if one can use the simplified version of pi for a d’Oraysa would be a disagreement between the Tosfos Rosh and the Rambam. According to the Tosfos Rosh (using the explanation of the Chazon Ish) all shiurim were given to Moshe at Sinai, and including that is the idea that when it comes to all mathematical equations one can use the simplified form. As the Torah was given to all, even to the simple minded people, the Torah allowed a simple calculation to be used. If this is the reason then even by a d’Oraysa one can use pi=3, as this was part of what was transmitted at Sinai.

However, the Rambam writes “But because this ratio is not precise and is only an approximation, they [the rabbis of the Mishnah and Talmud] used a more general value and said that any circle with a circumference of three has a diameter of one, and they used this value in all their Torah calculations” it seems that they only used it in their calculations and not in a mitzvah d’Oraysa.