User:Mohankumar Shetty/sandbox

Squaring a Circle – The Puzzle : Ancient geometers obviously posed the problem of Squaring the circle as they have had a belief that the problem could be solved only with an rational/algebraic value of Pi The only rational value of Pi ever used by ancient civilizations is the Babylonian value of 25/8@, which may be derived geometrically#. It is not known why this value did not reach or adopted by any other older civilizations. Modern mankind came to know of it only during the last century. Had Ancient geometers knew about this value of Pi, probably, “Squaring the Circle”; would not have remained an unsolved puzzle. The value of Pi in use is derived geometrically by measuring the perimeters of inscribed and super scribed polygons around a circle, but relying more on reasoning than logic. Archimedes might not have stopped with a 96 sided polygon for deriving the value of Pi; without certain logical reasoning and understanding of limitations of the method employed by him. With each increase in the number of sides, the apothem of the polygons approaches the length of the radius and hence, the ends of the sides of the inscribed polygon cannot remain within or on the circumference. With increase in number of sides, both the polygons tend to merge with each other. This value of Pi, if derived on the basis of area of the polygon with 96 sides; would be at variance with the value derived from the perimeter of the said polygon. It is therefore, an approximate value. It is subsequently proved that, constructing a square with the same area as a given circle by using only a finite number of steps; with compass and straightedge is an impossible task, since the Pi is a transcendental number. Therefore, this puzzle should be possible to be solved with Babylonian value of Pi. As per Babylonian value of Pi, area of a one unit diameter circle is 0.78125% of a unit square. Based on the fact that √3.125 ÷√2 = 1.25, the task is executed and the steps recorded in the following pages.