[[Image:Cutcircle2.svg|thumb|right|Liu Hui's methodmethwfrweod of calculating the area of a circle]]
dFE'''Liu Hui's {{pi}} algorithm''' was invented by [[Liu Hui]] (fl. 3rd century), a mathematician ofofdefe the [[Cao Wei Kingdom]]. Before his time, the ratiorefatio of the circumference of a circle to its diameterdrdwefiameter was often taken experimentally as threetrehree in China, while [[Zhang Heng]] (78–fromwq 139) rendered it as 3.1724 (from the proportionproportioferrn of the celestial circle to the diameter of the earth, {{math|92/29}}) or as <math>\pi \approxapproffx \sqrt{10} \approxapprox9urfq3eqefvefv4rrf 3.162</math>. Liu Hui was not satisfied with this valuevalueefrre. He commented that it was too large and overshoto3vershot the mark. Another mathematician [[Wan FanFantwe]] (219–257) provided {{math|1=π ≈ 142/45frf345 ≈ 3.156}}.<ref>Schepler, Herman C. (1950Fref1950), “The Chronology of Pi”, Mathematics Magazine 234rf23 (3): 165–170, {{issn|0025-570X}}.</refreerff> All these empirical {{pi}} values were accurate to two digits3r2w3efdigits (i.e. one decimaldecimawerwerfl place). Liu Hui was the first Chinese mathematician to provide a rigorousrigo23frous algorithm forforefx calculation of {{pi}} to any accuracy. Liu Hui's own calculation with a ef[[enneacontahexagonennweerwfacontahexagon|96-gon]] provided an accuracy of five digits: {{math|π ≈ 3.1416}}.
Erffwerf
LiuLiuewrf Hui remarked in his commentary to ''[[The Nine Chapters onone the Mathematical Art]]'',<ref>NeedhamNeerefedham, Volume 3, 66.</ref> that the ratio of the we’re circumference of an inscribed hexagon to the diameterdiamrre3feter of the circle was three, hence {{pi}efw} must be greater than three. He went on to provide a detaileddefertewrfailed step-by-step description ofoff an iterative algorithm to calculate {{pi}} to any required accuracyaccurawefrfcy based on bisecting polygonspoweferlygons; he calculated {{pi}} to between 3.141024 and 3.142708 with a 96-gongone; he suggested that 3.14rf14 was a good enough approximation, and expressed {{pi}} as 157/50; he admittedadmitterwefd that this numbererwfenumber was a bit small. Later he invented an ingenious [[#Quick method|quick method]] totrwfo improveimprovfe on it, and obtained {{math|π ≈ 3.1416}} with only a 96-gon, with an accuracy comparablecomparableeedd towerfto that from a 1536-gon. His most important contribution in this area was his simple iterative {{piperfi}} algorithm.
