Converting a square into an isosceles Triangle
The conversion of the square into an equivalent triangle is necessary for the construction of praugaciti. Prauga is a triangle part formed by the side beams and the axle of a cart. It has a shape of an isosceles triangle. And is explained by all three Baudhayana, Apastambha and Katyayana Shulbasutras.
Apstambha describes method as,
यावाग्निः सारत्निप्रादेशो द्विस्तावतीं भूमिं चतुरश्रां कृत्वा पूर्वस्याः करण्याः अर्धाच्छ्रोणी प्रत्यालिखेत्| सा नित्या प्रउगम्||(अपस्तम्भ शुल्बसुत्रम् १२.१०-११)
Having converted the area double that of the required fire altar measuring seven- and half purusha into a square, he should draw lines from the mid -point of the eastern side towards the western corners of the square. That is the exact prauga.
Let ABCD be a square of twice the required area. Let M be the midpoint of AB such that AM= MB, MD and MC are joined. Then MCD is the required triangle(prauga).
For if the altitude from MF is given, the square is divided into 2 equal rectangles AMFD and BMFC
Area of Triangle MDF = ½(rectangle AMFD)
Area of Triangle MFC = ½(rectangle BMFC
Then area of triangle MCD= ½(square ABCD)
This construction leads to the formula,
Area of a triangle = ½(base* altitude)