Π (pi) is an irrational number 3.14159... and it never ends. There is discussion in the literature (I haven't read any of it) whether 22/7 (=3.14285...) was the value of pi used by the Ancient Israelites and Greeks, which is not mathematically too close to the actual value.
There are at least two halachos in which Π plays an important role (excluding astronomy): Sukkah and Tefillin. In Sukkah the issue is the area of a [round] sukkah, and in Menachos the discussion is what is square, because our tefillin must be square. Recall the Pythagorean theorem which states that a^2 + b^2 = c^2, that is the square of the hypotenuse of a right triangle is equal to the sum of the square of the sides. That means that the length of the hypotenuse equals the square root of a^2+b^2. For a right triangle where the sides are 1 amah (or any unit), the hypotenuse equals the square root of 2=1.414. Critics claim, because the Gemara (Sukkah 8a; see also Shulchan Aruch O"C 32:39 from Menachos 35a) says that this hypotenuse equals 1.4 amos ("an amah and a fifth of an amah") that Chazal had an imprecise value of irrational numbers such as the square root of 2 and pi.
Dr. David Medved in his book "Hidden Light: Science Secrets of the Bible" quotes sources that Chazal had an extremely accurate measurement of pi; so accurate, in fact, that a more accurate value was not found for another 1000 years. While the vort he quotes, dealing with the word kav in Melachim and Divrei Hayamim, seems to be a cute vort but not pshat, I am willing to agree that Chazal may have had an accurate value, but they publicized a less accurate but far simpler value to use in calculations when needed.
Several prominent Torah personalities, such as Harav Chaim Zimmerman zt"l held that Chazal knew all scientific discoveries that would ever be found. Some of Rav Chaim Zimmerman's ideas on this topic might be found in his book "Torah and Reason," which I have not read, but I heard his views from several of his talmidim. (Rabbeinu Avraham ben HaRambam, in his famous essay on aggadah, found at the beginning of Ein Yaakov clearly disagrees with this view and says Chazal could have made mistakes in science.)
I believe I can prove that Chazal made use of less accurate values even when they knew of more accurate values, to keep things simple. When calculating the tekufos, the solar seasons, we have two values: the tekufa of Shmuel and the tekufa of Rav Adda. Shmuel holds (Eruvin 56a) each season lasts 91 days and 7.5 hours. Rav Adda holds the tekufa is 1 1/8 minutes shorter than Shmuel's tekufa. (See Rabbi J. David Bleich, "Bircas Hachammah," 1st edition, ArtScroll, p.49 for the calculation). It is a very involved calculation, especially compared to Shmuel's calculation.
Rav Bleich writes:
"This does not mean that Shmuel must have been ignorant of Rav Adda's method of calculation. Despite acceptance of Hillel's calendar based upon the divergent calculations of Rav Adda, Shmuel is depicted in the Gemara, Brachos 58b, as being familiar with the "paths of the sky" as he was with the alleys of his own city of Nehardea. It may be assumed that Shmuel adopted a simpler method of calculation in order to avoid the necessity of manipulating fractions. This is noted by so early an authority as R. Abraham Ibn Ezra who states in Sefer haIbbur, p. 8, that the tekufah of Shmuel is not the true tekufa, and, moreover, that Shmuel knew his announced calculations to be imprecise. Nevertheless, Shmuel chose a close approximation because of the difficulty which most people have in working in fractions. The same explanation is also advanced by the 17th-century Sephardic scholar, R. David Nieto in his Kuzari Sheni, Vikuach Chamishi, no. 146."
I feel they held people could use easy fractions like 2/5 and 1/4, but uglier fractions were more difficult to use.
Rav Bleich then brings another interesting and very logical possibility in regard to the use of Shmuel's tekufa in the calendar, but it is not so relevent to the simpler value of pi.