Wikipedia has a set of answers in their article on Approximations of pi. That links to a terrific article on rabbinic approximations of π by Boaz Tsaban and David Garber. Tsaban and Garber summarize as follows (pp. 10-11):
- The rational-religious approach of Maimonides holds that, since we cannot know the exact values, the Bible tells us that we do not have to worry about this and that is suffices to use the value 3.
- The mystical approach of [Matityahu Hacohen] Munk contends that 3 was indeed the ration of the circumference to the diameter in King Solomon's temple: This value is used in order to bridge the gap between our world and the "world of truth." For the sake of consistence, the halachic conditions are applied to the suitable regular polygons.
- The practical approach of R' Shimon Ben Tsemah [who learns from other places in Talmud that they used a more precise version of π] asserts the the rough approximations are used when teaching the students, but, when it comes to practice, the calculations are to be done by the experts.
So to answer your question, if you hold by Munk (I don't know who he is), then it's a miracle. If you hold by Rambam or R' Shimon ben Tsemah, it's an approximation
((Aside: two different psaks come out of this for practical reasons like sukkot - either you use the best mathematical approximation (R' Shimon ben Tsemah) or you use 3 (Rambam and Munk). In order to use 3 as π, you can just measure the perimeter of the interior inscribed regular hexagon.))