How Pi Was Derived and Relates to Area of a Circle

Date: 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/