SSJ Merilyn Ryan
10 April 2012
π- The History and Approximations One of the most interesting, and most commonly remembered, numbers in mathematics is the number π. It belongs to the irrational numbers, because it is a never ending decimal with no repeating pattern. In fact, it is one of the infinitely many irrational numbers. The value of π can only be written one way without using an approximation, ≈, symbol; and that way is the symbol π itself. Even with the help of super computers, we still do not know the exact value for π. So how did this value even come about, and make it so significant to the math world that it needed a name? It took the mind of one mathematician to even consider the thought of this value to be such an important number. Mathematicians knew that the perimeter of a circle was proportional to its diameter, which is twice its radius. This relationship can be written as
Perimeter of a Circle = dx = 2rx were d is the diameter, r is the radius, and x is the unknown value. They also knew that the area of the circle was proportional to the square of the radius. The relationship here can be expressed as
Area of a Circle = r2y with r still being the radius and y being the unknown (Slowbe). The equations have been observed and experimented with, but it was Archimedes who first constructed a proof that showed the two missing values, of these equations, are equivalent. In other words, Archimedes proved that x = y from the equations above (Slowbe). Today, we do not use the variable x or y; we use the constant value known as π. The equations are commonly written as
Perimeter of a Circle = 2πr
Area of a Circle = πr2 in mathematics today. Archimedes proved the equations above by doing calculations of the perimeters and areas of regular polygons inscribed and circumscribed about a circle (Slowbe). A regular polygon, as defined by Euclidean geometry, is a polygon with all the sides the same length and all the interior angles equivalent.
Now if we take regular polygons and inscribe them and circumscribe them about a circle, we notice something fascinating.
Inscribed Regular Polygon Number of Side Circumscribed Regular Polygon
Name of Regular Polygon
We notice that as we add more sides to the regular polygon, a shape starts to form similar to a circle. Archimedes noticed this as well, but he was the one who used it to change the world (Berengard). Archimedes started with a hexagon, then doubled the number of sides of the regular polygon multiple times, and used geometric equations to solve for the perimeters and areas of all the regular polygons. To help solve for the value of π, he wanted the solutions to be equal to the value desired; meaning he set the perimeter and the area of the circle equal to π. Since he wanted these results, he had to use two different circles, one for the area and one for the perimeter, since the area of a circle is not equal to its perimeter. With calculations, he noticed that he should make the radius of the circle equal to ½ (r=0.5) to solve for the perimeter to be equivalent to π. He also notice that he should make the radius equal to 1 (r=1) to solve for the area to be equivalent to π (Perimeter) = 2πr = π (Area) = πr2 = π ½π(2πr) =½π(π) 1/π(πr2) = 1/π(π) r = 1/2 r2 = 1 r = 1 Archimedes used the two different radii of the two circles, to come up with a few equations for the perimeter and the area of the regular polygon, using Euclidean geometry.
Perimeter of an Inscribed n-gon Perimeter of a Circumscribed n-gon
Area of an Inscribed n-gon
Area of a Circumscribed n-gon
(Slowbe). After Archimedes had his equations, he replaced n with the number of sides of the regular polygon into the equations. He was then able to get intervals, for the perimeter and area of