The Pythagorean Theorem - Educational Outreach

Figure 4.16: Triangular Phi---The history of PI, the fixed ratio of the circumferenceto the diameter for any circle, parallels the history of thePythagorean Theorem itself. PI is denoted by the Greekletter . Like the Golden Ratio Phi, PI is a topic of sufficientmathematical weight to warrant a complete book in its ownright. As one might suspect, books on PI have already beenwritten. In this volume, we will simply guide the reader tothe two excellent ‘PI works’ listed in the References.In this Section, we will illustrate just one of manydifferent ways of computing Pi to any desired degree ofaccuracy. We will do so by way of a nested iterativetechnique that hinges upon repeated use of the PythagoreanTheorem.Figure 4.17 on the next page shows a unit circlewhose circumference is given by C 2. Let T1be anisosceles right triangle inscribed in the first quadrant whosehypotenuse is h 12 . If we inscribe three right trianglescongruent to T1, one per each remaining quadrant, then acrude approximation to C is given by 4h 1 4 2 .160