Why did Archimedes use a 96-gon to estimate the value of pi?

And to add to what David said, as for why you start that series at n=6, well, to get the area of the hexagon, it divides into 6 equilateral triangles with side lengths of r, the radius of the circle. So because it's equilateral, you get the base, b, of the triangle is r, and the height is reasonable to get from Pythagoras, giving h=sqrt(3)/2. read more

And to add to what David said, as for why you start that series at n=6, well, to get the area of the hexagon, it divides into 6 equilateral triangles with side lengths of r, the radius of the circle. read more

Archimedes' Approximation of Pi One of the major contributions Archimedes made to mathematics was his method for approximating the value of pi. It had long been recognized that the ratio of the circumference of a circle to its diameter was constant, and a number of approximations had been given up to that point in time by the Babylonians, Egyptians, and even the Chinese. read more

Much earlier, at least 1500 years before Euclid, the ancient Egyptians had an estimate of this constant of proportionality, and their estimate was about 3.16. What Archimedes did was carry out the process in Euclid's XII.2 using regular 96-sided polygons. read more

Most online explanations of Archimedes procedure cheat and use trigonometry but here we'll do it the way Archimedes did it using only algebra and geometry. The English translation of Proposition 3 in"The Measure of a Circle" used as my source and is from"The Works of Archimedes" translated from Greek by T. L. Heath and published in 1897. read more