Dr. Ashish Karn
Converting square to circle and circle to square
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 northeast corner towards the eastwest 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 onethird 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 onethird 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.

Area of ABCD = 4 square units

Area of MNOP = 2 square units

Area of inscribing circle= 3 square units approximately ( as it lies between ABCD and MNOP)

The area of subscribing circle = 6 square units

“The area of a circle describing on the diagonal of a square is double that of the circle described on the side of the square” (Maitrayana Shulbasutra 1.1.8)

As the area of the required circle is supposed to be equal to that of a given square is 4 square units. The required circle will lie somewhere between the subscribed circle and the inscribed circle.

Dividing the difference of the radii of the circles into three parts such that MQ = QR = RS = 1/3 MS.

The radii EQ and ER will describe the circles having an area approximately equal to 4 and 5 square units respectively. Hence, It is clear that Shulabkaras might have come to the conclusion that the third part of the radius lying outside the square taken with half of the side of the square will be the radius of the required circle.