The rope which is stretched across the diagonal of a square produces an area double the size of the original square.The Katyayana Sulbasutra however, gives a more general version:-
The rope which is stretched along the length of the diagonal of a rectangle produces an area which the vertical and horizontal sides make together.The diagram on the right illustrates this result.
Note here that the results are stated in terms of “ropes”. In fact, although sulbasutras originally meant rules governing religious rites, sutras came to mean a rope for measuring an altar. While thinking of explicit statements of Pythagoras‘s theorem, we should note that as it is used frequently there are many examples of Pythagorean triples in the Sulbasutras. For example (5, 12, 13), (12, 16, 20), (8, 15, 17), (15, 20, 25), (12, 35, 37), (15, 36, 39), (5/2 , 6, 13/2), and (15/2 , 10, 25/2) all occur. Now the Sulbasutras are really construction manuals for geometric shapes such as squares, circles, rectangles, etc. and we illustrate this with some examples. The first construction we examine occurs in most of the different Sulbasutras. It is a construction, based on Pythagoras‘s theorem, for making a square equal in area to two given unequal squares. Consider the diagram on the right. ABCD and PQRS are the two given squares. Mark a point X on PQ so that PX is equal to AB. Then the square on SX has area equal to the sum of the areas of the squares ABCD and PQRS. This follows from Pythagoras‘s theorem since SX2 = PX2 + PS2. The next construction which we examine is that to find a square equal in area to a given rectangle. We give the version as it appears in the Baudhayana Sulbasutra. Consider the diagram on the right. The rectangle ABCD is given. Let L be marked on AD so that AL = AB. Then complete the square ABML. Now bisect LD at X and divide the rectangle LMCD into two equal rectangles with the line XY. Now move the rectangle XYCD to the position MBQN. Complete the square AQPX. Now the square we have just constructed is not the one we require and a little more work is needed to complete the work. Rotate PQ about Q so that it touches BY at R. Then QP = QR and we see that this is an ideal “rope” construction. Now draw RE parallel to YP and complete the square QEFG. This is the required square equal to the given rectangle ABCD. The Baudhayana Sulbasutra offers no proof of this result (or any other for that matter) but we can see that it is true by using Pythagoras‘s theorem.
It is worth noting that many different values of π appear in the Sulbasutras, even several different ones in the same text. This is not surprising for whenever an approximate construction is given some value of π is implied. The authors thought in terms of approximate constructions, not in terms of exact constructions with π but only having an approximate value for it. For example in the Baudhayana Sulbasutra, as well as the value of 676/225, there appears 900/289 and 1156/361. In different Sulbasutras the values 2.99, 3.00, 3.004, 3.029, 3.047, 3.088, 3.1141, 3.16049 and 3.2022 can all be found; see . In  the value π = 25/8 = 3.125 is found in the Manava Sulbasutras. In  in addition to examining the problem of squaring the circle as given by Apastamba, the authors examine the problem of dividing a segment into seven equal parts which occurs in the same Sulbasutra. The Sulbasutras also examine the converse problem of finding a circle equal in area to a given square. Consider the diagram on the right. The following construction appears. Given a square ABCD find the centre O. Rotate OD to position OE where OE passes through the midpoint P of the side of the square DC. Let Q be the point on PE such that PQ is one third of PE. The required circle has centre O and radius OQ. Again it is worth calculating what value of π this implies to get a feel for how accurate the construction is. Now if the square has side 2a then the radius of the circle is r where
Increase a unit length by its third and this third by its own fourth less the thirty-fourth part of that fourth.Now this gives