Plane Geometry - Bruce E. Shapiro

Section 44Estimating πInscribed PolygonsWhile there are many ways of calculating π using calculus (e.g., as the sumof an infinite series), here we outline a procedure based purely on geometriccalculations. It is iterative, in the sense that we need to calculate a sequenceof estimates based on previous estimates, slowly converging to the desiredvalue. We will define π n as the ratio of the perimeter to the diameter ofa succession of inscribed polygons with 2 n sides, starting with n = 2. Theidea is that as more sides are added the inscribed polygon looks more andmore like a circle (rigorously, the limit of the perimeters is the circumferenceof the circle). In general, the perimeter of the n th polygon isP n = 2 n H nwhere H n is the length of the outer edge. Since the diameter of the unitcircle is always 2, we have an estimate for π ofπ n = 1 2 P n = 2 n−1 H nFor n = 2, we inscribe a square in a circle of radius 1, and divide the squareinto four right triangles with hypotenuse H 2 .ThenH 2 = √ 2π 2 = 2 2−1 H 2 = 2 √ 2 ≈ 2.82843For n = 3 we divide each central angle in half. Define the distance a asshown in figure 44.2.239