Last June π was calculated to 202 trillion digits based on this hypergeometric equation.Since π is an irrational number with an infinite number of decimal places, a geometrical model representation as found in Kings will by definition be an approximation.
Similarly the Babylonians, Egyptians, Persians, Greeks, Indians and Chinese used geometrical models to calculate π to various degrees of accuracy.
Although the Greeks introduced the idea of the irrational number, π was proven to be an irrational number in the 18th century.
In the 14th century Indian mathematicians who were centuries ahead of their Western European counterparts in the field of trigonometry found inverse trigonometric functions such as arctan(x) could be expressed as a series expansion from which π could be expressed as the series.
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This series was found by the Indian mathematician Madhava in the 14th century and independently discovered by Isaac Newton’s great rival Gottfried Leibnitz three hundred years later.
By summing more terms in series resulted in a greater accuracy of π.
When π was found to be irrational the series was modified to the following.
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What this equation states π is a limit, since the number of terms summed can only be finite the more terms summed the closer the limit is reached without ever attaining it.
This is illustrated by a computer code I wrote up for the Madhava-Leibnitz series summing it from 1 term to 20000 terms in increments of 10.
Note the summed terms will never reach the value of π irrespective of how many terms I used to sum the series.

They did it again: Tech publisher keeps on breaking Pi Calculation World Record — they almost doubled the previous one, reaching 202 trillion digits in 100 days and used 1.5PB of SSD storage
Previous record of 105 trillion digits broken again
The difference between the 14th century equation to calculate π and this late 20th century equation is that while it took around 20000 iterations for the calculated value of π to converge to a value reasonably close to the 'true value' of π, it took only 1 iteration for the computer code I wrote up with the 20th century equation.
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