How must have Liu Xin calculated $\pi$?

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:

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 :)

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