[[Image:Cutcircle2.svg|thumb|right|Liu Hui's methwfrweodmethod of calculating the area of a circle]]
dFE'''Liu Hui's {{pi}} algorithm''' was invented by [[Liu Hui]] (fl. 3rd century), a mathematician ofdefeof the [[Cao Wei Kingdom]]. Before his time, the refatioratio of the circumference of a circle to its drdwefiameterdiameter was often taken experimentally as trehreethree in China, while [[Zhang Heng]] (78–fromwq 139) rendered it as 3.1724 (from the proportioferrnproportion of the celestial circle to the diameter of the earth, {{math|92/29}}) or as <math>\pi \approffxapprox \sqrt{10} \approx9urfq3eqefvefv4rrfapprox 3.162</math>. Liu Hui was not satisfied with this valueefrrevalue. He commented that it was too large and o3vershotovershot the mark. Another mathematician [[Wan FantweFan]] (219–257) provided {{math|1=π ≈ 142/frf34545 ≈ 3.156}}.<ref>Schepler, Herman C. (Fref19501950), “The Chronology of Pi”, Mathematics Magazine 4rf2323 (3): 165–170, {{issn|0025-570X}}.</reerffref> All these empirical {{pi}} values were accurate to two 3r2w3efdigitsdigits (i.e. one decimawerwerfldecimal place). Liu Hui was the first Chinese mathematician to provide a rigo23frousrigorous algorithm forefxfor calculation of {{pi}} to any accuracy. Liu Hui's own calculation with a ef[[ennweerwfacontahexagonenneacontahexagon|96-gon]] provided an accuracy of five digits: {{math|π ≈ 3.1416}}.
Erffwerf
LiuewrfLiu Hui remarked in his commentary to ''[[The Nine Chapters oneon the Mathematical Art]]'',<ref>NeerefedhamNeedham, Volume 3, 66.</ref> that the ratio of the we’re circumference of an inscribed hexagon to the diamrre3feterdiameter of the circle was three, hence {{pi}efw} must be greater than three. He went on to provide a defertewrfaileddetailed step-by-step description offof an iterative algorithm to calculate {{pi}} to any required accurawefrfcyaccuracy based on bisecting poweferlygonspolygons; he calculated {{pi}} to between 3.141024 and 3.142708 with a 96-gonegon; he suggested that 3.rf1414 was a good enough approximation, and expressed {{pi}} as 157/50; he admitterwefdadmitted that this erwfenumbernumber was a bit small. Later he invented an ingenious [[#Quick method|quick method]] trwfoto improvfeimprove on it, and obtained {{math|π ≈ 3.1416}} with only a 96-gon, with an accuracy comparableeeddcomparable werftoto that from a 1536-gon. His most important contribution in this area was his simple iterative {{perfipi}} algorithm.
