How Aryabhatta approximated value of PI

Following is an excerpt from a research article titled "Best k-digit rational approximation of irrational numbers: Pre-computer versus computer era" by S.K. Sena, and Ravi P. Agarwal
Department of Mathematical Sciences, Florida Institute of Technology

More than 4700 years ago, the famous Indian mathematician and astronomer Aryabhatta (b. 2765 BC) gave 62832/20000 = 31416/10000 = 3.1416 as an approximation of π [21]. He calculated π by measuring the diameter of the circle in a remainderless unit and then measuring the circumference in the same unit. He made the units increasingly smaller in such a way that the diameter would be an integer in the unit used and improved the accuracy [21]. It is interesting to note that the two parameters, viz., (i) the size of the circle and (ii) the unit employed will be vital to improve the accuracy of π. By increasing the size of the circle and keeping the unit of measurement fixed or by making the unit size smaller and keeping the circle fixed, one can improve accuracy significantly. Thus keeping the circle as large as possible within the limits of the concerned measuring device and then making the unit of measurement as small as possible, one can achieve the maximum possible accuracy. It is interesting to note that there is, in general, no measuring device – optical or electronic or any other based on any other technology – that can measure a quantity with an accuracy more than 0.005% [3]. This translates to four significant digits. Thus, even over 4700 years ago when measuring devices were believed to be less sophisticated, Aryabhatta obtained π(=3.1416) to an accuracy of almost four digits! Beyond the accuracy of four significant digits, it is not possible to compute π (by measuring the circumference and the diameter of a circle) even today when much more sophisticated measuring devices are available! This is definitely a remarkable achievement by Aryabhatta in mathematics in ancient India! Aryabhatta also discovered the non-remainderlessness of the circumference of a circle when the diameter is measured in a unit which provides an exact integer (within the limits of device error) for the value of the diameter [21]. This fact clearly demonstrates that Aryabhatta and for that matter the then Indian mathematicians/astronomers knew that π is an irrational number, however, finitely small the unit of measurement be. Another Indian mathematician, Bhaskara (who was in between Aryabhatta (2675 BC) and another famous Indian astronomer Varahamihira (123 BC) and whose exact time is not known) was the earliest known commentator of Aryabhatta’s works [21]. He suggested several approximations for π − 3927/1250 (=3.1416) for accurate work, 22/7 (=3.142857142857143) for less accurate calculation, while (=3.162277660168380) for ordinary work [1].