Liu Hui used the area of n-gon inscribed in a circle (with n being the larger the better) to estimate the area of a circle. In his own word, he said that “The finer the circle being cut, the smaller error we get when we estimate; if we continue to cut until the polygon can’t be cut any more, then the area of this polygon yields the area of the circle”. It is worth noticing that his method coincides with the modern infinite series methodology of estimation. Basically, his theory is to say that : When n goes to infinity, n-gon is congruent to circle.
(Liu Hui’s approximation of pi using n-gon)
From Qin Dynasty (221 B.C.E.), the value of pi (‘ratio of perimeter and diameter’ in Chinese thought) was generally believed to be 3, until 張衡 (Zhang Heng, 78-139 C.E., ancient Chinese mathematician, astronomer and geologist) questioned the accuracy of this value and calculated pi to be sqrt10, and after which Liu Hui suggested that using pi = 3 the perimeter we calculated is actually the perimeter of hexagon inscribed in the circle, instead of the circle itself, which was an underestimation. Subsequently, Liu attempted the value sqrt10 and found that this value tends to be an overestimation. Thus Liu Hui began to cut the circle from a hexagon, then by cutting every side of the hexagon into halves could we get a dodecagon, similarly could we get 24-gon (basically 2^n*6-gon), this went on and on till a 3072(2^9*6)-gon was got and finally he calculated a comparatively precise value of pi to be 3.1416.
(Details about Liu’s pi algorithm)
The image below is an example of Liu’s method of calculating the area of 2n-gon inscribed in a circle.
Basically we have already calculated the area of n-gon before, and the area of triangle AOB is 1/n of the area of the n-gon. For the next ‘cut’, we cut side AB to get two equal sides, namely AC and BC. So in order to know the new area of the 2n-gon, all we need to know is the area of the newly existed small triangle ACG (or CGB); from which we would know the area of triangles AOC and BOC.
We know that OA = OB = OC = r, while AB (side length of the n-gon, which is known) = l.
We have that AG = GB = l/2, and that OG = sqrt(BO^2-BG^2) = sqrt(r^2-l^2/4).
Thus CG = OC – OG = r – sqrt(r^2-l^2/4).
So the area of triangle ABC is S = 1/2 bh = 1/2AB*CG = BG*CG = l/2 * (r – sqrt(r^2-l^2/4)).
Hence, the area of the 2n-gon is the area of the former n-gon plus n times the area of S. We could start with a hexagon where l = r. The farther we calculated the smaller error we have.
(sculpture of Zu Chongzhi in China)
A better approximation of pi using Liu’s algorithm was calculated by 祖沖之 (Tsu Chung-Chi, 429-500 C.E., ancient Chinese mathematician) several hundred years later. Tsu used Liu’s method and was the first person to have approximated the value of pi to the 7th decimal places (between 3.1415926 and 3.1415927) using a 24576(2^12*6)-gon. He calculate both an underestimation and an overestimation of pi. For the latter one, he probably used the area of the former n-gob plus 2n times the area of S to overestimate the area of the circle.
For us who luckily live in the modern society, we could calculate the limit of the area of n-gon when n goes to infinity to get an exact expression of pi.