From: "Richard Guy" <rkg@cpsc.ucalgary.ca>
Surely best to follow Archimedes and calculate the perimeters of inscribed and circumscribed regular 6-, 12-, 24-, ...-gons? R.
Another approach is using a polynomial approximation of sin. Linear approximation sin(x) = x is not very good. For x=pi/6 it gives 1/2=pi/6 and pi=3. However, cubic approximation sin(x) = x + Ax^3 is already pretty good (quadratic term is not present because sin is odd). From sin(pi/6)=1/2 and sin(pi/4)=sqrt(2)/2=.707 we get a system of linear equations pi/6 + A pi^3/6^3 = 0.5 pi/4 + A pi^3/4^3 = 0.707 Multyplying first equation by 6^3 and second - by 4^3, we get 36 pi + A pi^3 = 108 16 pi + A pi^3 = 45.2 Subtracting, 20 pi = 62.8. and pi = 3.14 Alec Mihailovs