Zhao Youqin'sπ algorithm is an algorithm devised byYuan dynasty Chinese astronomer and mathematicianZhao Youqin (赵友钦, ? – 1330) to calculate the value ofπ in his bookGe Xiang Xin Shu (革象新书).

Zhao Youqin started with an inscribed square in a circle withradiusr.^[1]

If${\displaystyle \ell }$ denotes the length of a side of the square, draw aperpendicular lined from the center of the circle to sidel. Lete denotesr − 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 lined to dissect the circle into anoctagon;${\displaystyle {\ell }_2}$ 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 ofhexadecagon

${\displaystyle {\ell }_3=\frac{1}{2}\sqrt{{\ell }_2^2+4{\left(r-\frac{1}{2}\sqrt{4r^2-{\ell }_2^2}\right)}^2}}$

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

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

He multiplied this number by 113 and obtained 355. From this he deduced that of the traditional values ofπ, that is 3, 3.14,⁠22/7⁠ and⁠355/113⁠, the last is the most exact.^[2]

• Liu Hui'sπ algorithm

• Chinese mathematical discoveries
• Pi algorithms

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