Why Maths?

Approximating Pi The number Pi (?) is a mathematical constant that is the ratio of a circle`s circumference to its diameter, and is approximately equal to 3.14159. It is the same number no matter the circle you use to compute it. Research proves that the mathematical phenomena Pi (?) dates as far back as the bible. A man called Archimedes of Syracuse (281-212BC) was first credited with the theoretical calculation of Pi. Using polygons whose sides were tangent to a circle, known as circumscribed polygons he was able to prove that Pi was between 3.10 and 3.14. Pi is also an irrational number, that is to say its decimal representation never ends and it never settles into a permanent repeating pattern. ( Maths.com, 2005). For the purpose of this task, I will be looking at Archimedes' method of approximating Pi. Using Geometry Sketch Pad (GSP) I started with constructing a hexagon placed inside a circle so that all the vertices of the polygon lie on the circumference of the circle, otherwise termed an inscribed hexagon. I calculated the perimeter and used the ratio of the perimeter to the diameter of the circle as an estimate of Pi. Subsequently, I doubled the sides to form a dodecagon and then a 24 sided polygon. I applied the process above to a square, and created an animation of the polygons to demonstrate the process and effect on GSP. I observed that the ratio of the diagonal, which is also the diameter of the circle to the polygon, increases as the sides of the polygon increase and moves closer to the circle. This ratio never goes above 3.14, so as the sides are increased the ratio moves towards Pi but never goes above it. Also, translating the shape, the perimeter and diameter varies while this ratio remains constant. Though GSP was demonstrative and interactive, it was also restrictive as I could only go up to a 60 sided polygon and I was unable to increase the decimal places. Microsoft Excel lent itself better for this purpose as it returns Pi accurate to 15 digits. My conjecture therefore was that as the number of sides of the polygon increases, the ratio of the perimeter to the diameter gets closer to Pi but never goes above Pi. Employing my knowledge of polygons, I went ahead to test my conjecture and then created a model which I put into Excel.

. Importing the formulas above into Excel: The Excel computation above substantiates my conjecture that the ratio of the perimeter/ diameter truly gets closer to Pi as you increase the number of sides of the polygon. At 12 sides the ratio is equal to Pi correct to one decimal Place; at 96 sides it is correct to two decimal places. From 6144 sides notice how the ratio remains constant at 3.141593. This demonstrates that Pi (Circumference/ diameter) is always 3.14159. To use this formula I had to keep doubling the sides of the polygon, this meant that the polygon had to have an even number of sides. Our tutor, Richard Burnett had once mentioned that using the area of polygons in approximating Pi allows you to use polygons with odd or even number of sides. So I decided to try it out as an extension, I have included this in my portfolio. In carrying out this task, Excel facilitated a more exhaustive demonstration. Archimedes was constrained by hand calculations in the number of sides of polygons he used. ICT on the other hand, has made it possible to increase the number of sides of the polygon almost infinitely, thereby improving on Archimedes' work. Prior to this task I have never quite understood the notion of Pi. Following the visualization and understanding the task has afforded me, I am confident that I won't simply be using Pi anymore; rather I will recognize why I am using it. Furthermore, I now possess a deeper understanding of 2D shapes and their properties. However, I would like to work on my competence in the use of GSP, and I am thinking about taking this on as a project.