The early history of converting a square into a circle and a circle into a square  dates back to the time of Rigveda (before 3000B.C.). That was in the connection with the construction of three primary essential sacrificial altars namely, Garhapatya, Ahavaniya, and Dakshinagni altars. These three altars had to be of the same areas but of different shapes, the first circular, the second square, and the last semicircular, the archaic variety of Samashana citi is stated to be circular, Further instances of the early application of the above geometrical operation are found in taittriya and other Samhita in connection with the construction of Rathachakra citi, Samuhya citi, parichaya citi, and Drona citi. In each of these cases one has to draw at first a square equal in area to that of the primitive syena citi and later square has to be circled.  Baudhayana and Apastambha clearly describe these methods of converting squares to circles and vice versa.

Square to circle:

Baudhayana shulbasutra describes methods of converting square to circle as:

चतुरस्त्र मंडलं चिकीर्षन्नक्ष्णयार्ध मध्यात्प्राचीमभ्यापातयेत् यदतिशिष्यते तस्य सह तृतीयेन मंडलं परिलिखेत्|| |बौधायनशुल्बसुत्रम्  १.५८||

"If you wish to transform a square into a circle, at the center of the square, fix acord equal to half of the diagonal of the square and draw an arc from the north-east corner towards the east-west line; draw a circle together with the third part lying outside the square". (B. S. 1.58)

Similarly, Katyayana shulbasutra describes method stated as

चतुरस्त्र मंडलं चिकीर्षन्मध्यादंसे निपात्य पार्श्वतः परिलिख्य तत्र यदतिरिक्तं भवति, तस्यत तृतीयेन सह मंडलं परिलिख्य समाधिः|| कात्यायन शुल्बसुत्रम्   ३.१३||

"If you wish to transform a square into a circle, stretch the cord from the center to the amsha (the east–west or the south–north corner) and draw an arc towards the side of the square. Then with half of the side together with the one-third potion that lies outside the square draw a circle. That is the method".(K. S. 3.13)

For converting a square into a circle,

• let us construct a square ABCD having each side equal to 2a, with O as center and OA as the radius draw an arc intersecting the extended line OE (OE = half of the side of the square) at F.

• Divide EF into three equal parts such that EG = 1/3 EF. Then with center O, and radius OG describes a circle. This is the required circle which is approximately equal to the square. 

Now, what is the rationale for dividing the portion lying outside the square into three parts and taking one-third with half of the side of the square for drawing a circle of the required area?

•  Katyayana shulbasutra clearly elaborates on the rationale of the formula for converting a square to a circle, According to it, Let ABCD  be a square having each side equal to 2 units and E be the center of the square. M, N, O, and P be the midpoints of AB, BC, CD, and DA respectively.  Draw square MNOP and circle inscribing the square ABCD and circle subscribing it.