Constructing π from a regular 12-gon
Just approximately, of course! It is impossible to create the exact length π from the unit length by using a compass and a straightedge, since π is a transcendental number, and only (certain) algebraic numbers can be constructed in that way.
In this figure all sides of the regular 12-gon are 1 unit long. By following the figure, the blue length is very close to π, its accuracy is 4 digits.
In fact, its length is the same as the one can be obtained by using Kochański's construction (1685).
The presented construction was found by the RegularNGons program with the input n=12, case 115961. (This case is a symmetric variant of case 52958.) Other similar constructions can be found for regular star-12-gons as well.
It seems very likely that no better approximation can be found in a similar way, if the number of the sides is small, only by using regular (star-)12-gons. The best approximation among the regular (star-)10-gons is 3.14134, and among the regular (star-) 8-gons is 3.13503 (both are constructible). Kochański's approximation, however, outperforms both of them.