How Pi Was Derived and Relates to Area of a CircleDate: 03/04/98 at 15:58:17 From: Manuel Fregoso Subject: geometry(circles and pi) I would like to know how the value of pi was derived and how it relates to the area of the circle. Date: 03/04/98 at 16:56:50 From: Doctor Rob Subject: Re: geometry(circles and pi) The ancient Greeks knew that the ratio of the circumference of a circle to its diameter is a constant, which we call Pi. Archimedes, in particular, calculated this constant by computing the perimeter of a regular n-gon inscribed in the circle, and one circumscribed around the circle. The inscribed polygon has a smaller perimeter than the circumference of the circle, and the circumscribed one has a larger one. For him, n = 3, 6, 12, 24, 48, and 96. In this way, Archimedes proved that 3 + (10/71) < Pi < 3 + (1/7), which is not bad! Similarly, the area of the circle is between the areas of the two polygons. It turned out that the ratio of the area of the circle with radius r to the area of a square of side r is also the same constant, Pi. This means that the area of a circle of radius r is A = Pi*r^2. One way to see this is to look at the inside and outside n-gons, and cut them into triangles. Each triangle will have one vertex at the center, and the other two vertices will be adjacent vertices of the n- gon. The area of each triangle is half its height times the length of the base. The area of all the triangles together is half their height (they all have the same height because the n-gon is regular) times the sum of the lengths of their bases. This last sum is just the perimeter of the n-gon. For the outside n-gon, the height is just the radius of the circle. For the inside n-gon, as the number of sides increases, the height approaches the radius of the circle. For both, the perimeter of the n-gon approaches the circumference of the circle. Thus, the total area of each n-gon approaches the same number, half the radius times the circumference, or (1/2)*r*(2*Pi*r) = Pi*r^2. This must be the area of the circle. Nowadays, we prove that the two ratios are the same using calculus, where it is a simple exercise. Furthermore, we have much more effective algorithms for computing Pi based on ideas developed in the last two centuries. Recently, with the help of powerful digital computers, Pi has been computed to over 51,000,000,000 decimal places (that's 51 billion), an astonishing feat! -Doctor Rob, The Math Forum Check out our web site http://mathforum.org/dr.math/ |
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