Zhao Youqin's π algorithm

Zhao Youqin's π algorithm is an algorithm devised by Yuan dynasty Chinese astronomer and mathematician Zhao Youqin (赵友钦, ? – 1330) to calculate the value of π in his book Ge Xiang Xin Shu (革象新书).

Zhao Youqin started with an inscribed square in a circle with radius r.

If ${\displaystyle \ell }$denotes the length of a side of the square, draw a perpendicular line d from the center of the circle to side l. Let e denotes r − d. Then from the diagram:

${\displaystyle d={\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}}$

${\displaystyle e=r-d=r-{\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}.}$

Extend the perpendicular line d to dissect the circle into an octagon; ${\displaystyle \ell _{2}}$denotes the length of one side of octagon.

${\displaystyle \ell _{2}={\sqrt {\left({\frac {\ell }{2}}\right)^{2}+e^{2}}}}$

${\displaystyle \ell _{2}={\frac {1}{2}}{\sqrt {\ell ^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell ^{2}}}\right)^{2}}}}$

Let ${\displaystyle l_{3}}$denotes the length of a side of hexadecagon

${\displaystyle \ell _{3}={\frac {1}{2}}{\sqrt {\ell _{2}^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell _{2}^{2}}}\right)^{2}}}}$

similarly

${\displaystyle \ell _{n+1}={\frac {1}{2}}{\sqrt {\ell _{n}^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell _{n}^{2}}}\right)^{2}}}}$

Proceeding in this way, he at last calculated the side of a 16384-gon, multiplying it by 16384 to obtain 3141.592 for a circle with diameter = 1000 units, or

${\displaystyle \pi =3.141592.\,}$

He multiplied this number by 113 and obtained 355. From this he deduced that of the traditional values of π, that is 3, 3.14,

• Liu Hui's π algorithm