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 :)