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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)

Tool 3_Construct an isosceles triangle equal to a square.png
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