Circle into a square
The method of converting a circle into a square has been stated by Baudhayana, Apastambha, and Katyayana Shulbasutra. Baudhayana Shulbasutra stated two methods of converting circle to square first method is stated as,
मण्डलं चतुरस्त्रं चिकीर्षन् विष्कम्भमष्टौ भागान्कृत्वा भागमेकोनत्रिंशधा विभज्याष्टामंडलंविंशतिभागानुद्धरेद् भागस्य च षष्ठमष्टमभागोनम् ||बौधायनशुल्बसुत्रम् १.५९||
“If you wish to transform a circle into a square, having divided the diameter into eight parts and one of these eight parts into twenty-nine parts, remove twenty-eight parts; remove the sixth part of the remaining one part less the eight part of the sixth part.” (B.S 1.59)
Another method is given as
अपि वा पञ्चदश भागान्कृत्वाद्वावुद्धरेत् सैशानित्या चतुरस्त्रकरणी||बौधायन शुल्बसुत्रम् १.६०||
“Or else, divide the diameter into fifteen parts and remove two parts; the side of the square, thus obtained, is approximate.”
According to this method, the value of the side of a required square can be obtained by dividing the diameter of the given circle into fifteen parts and removing the two. And It has been explicitly admitted by the authors of sulbasutra that it yields gross(anitya) value.
That is to say, Let 2a be the side of the required square and d be the diameter of the given circle.
Then, 2a = d – (2/15) d
Or a = r- (2/15) r
The logic underlying the formula can be explained if it is assumed that the Shulbakaras have the knowledge of the value of √3 and that is 26/15 approximately.
Draw a square ABCD circumscribing the circle and also the square A'B'C'D' inscribed within it. Then apparently the area of the circle will be smaller than the area of the square ABCD (= 4r ) and greater than the area of the square A'B'C'D' (=2r ); that is
4r > Area of circle > 2 r .
The area of the circle will be 4r + 2 r /2 = 3 r .
If 2a denotes aside of square equivalent in area to the circle, we shall have approximately,
4a = 3r
Or a = √3 r/2
If √3 = 26/15 then a = r- 2r/15
Geo-gebra computational tool to convert circle to square