I know there won't be a definite answer to this question because of a lack of required historical evidence but there is so much we know of Liu Xin and of that time period. In that time, to get to an approximation 3.154 was a phenomenal feat.

To this day, no one knows how Liu Xin did it but we can speculate, can't we?

According to Science and Civilisation in China, Volume 3: Mathematics and the Sciences of the Heavens and the Earth by Joseph Needham, we do not know how Liu Xin (Liu Hsin) arrived at this approximation for $\pi$

On page $100$ of the book, Needham writes the following: xin

If you can find the records of how he did it, that would be great. But I ask this soft-question so that someone can correctly guess how it was done. I'm sure someone will formulate the same method. Liu Xin was also human :)

  • $\begingroup$ When did Liu Xin live ? I guess long before Archimedes because Archimedes already calculated $\pi$ much more precisely. $\endgroup$ – Peter May 25 '17 at 11:26
  • $\begingroup$ @Peter No, he was a contemporary to Augustus, so 200 years after Archimedes' death. $\endgroup$ – user228113 May 25 '17 at 12:05
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    $\begingroup$ Maybe a question for hsm.stackexchange.com. $\endgroup$ – Martín-Blas Pérez Pinilla May 26 '17 at 8:54
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    $\begingroup$ Could it be that Liu Xin had no theory about π, but simply based his figures on measurements? $\endgroup$ – PJTraill Aug 6 '18 at 13:01
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    $\begingroup$ @Nick: It would be a little unfortunate, I suppose! But one has to bear the possibility in mind. $\endgroup$ – PJTraill Aug 6 '18 at 15:41

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