Calculation of an Approximation to Pi

Pi is irrational and transcendental (that is, it cannot be calculated to a finite value). It is possible however to represent Pi using mathematical models, generally geometric in origin and often resolving into an infinite series or a repeated loop spiraling into infinity. Such methods would require an infinite number of steps to provide a finite solution, however they make it possible to calculate an approximate value for Pi, the accuracy of which is limited only by the time and patience applied to the problem. Each step must be calculated to the degree of accuracy required for the solution and in the absence of modern tools (calculators and computers) only a limited number of decimal places could be obtained. Over the millennia results have been obtained with increasing accuracy and with the advantage of modern technology a Japanese team, Tamura and Kanada, obtained Pi to almost 17 million decimal places. The method was based on Gauss's study of the arithmetic and geometric mean of two numbers and made use of a super computer (with multiple processors).
'The Penguin Dictionary of Curious and Interesting Numbers' (by David Wells) devotes several pages to the history of attempts to evaluate Pi. Reading this made me wonder what could be achieved using a desktop computer. Microsoft Excel lends itself to repeated calculations so carrying out the exercise in a spreadsheet would reduce the amount of manual effort involved.
I reasoned that a circle could be defined as a uniform polygon with an infinite number of sides. As the number of sides tends towards infinity the length of the sides tends to zero and the periphery of the polygon tends towards the circumference of a circle. (Sounds a bit like Calculus?). My method began with a polygon of known periphery, a hexagon. Equating this to a circle gives Pi = 3. I inserted this in the top row of the worksheet as starting data and used Pythagoras to calculate the data for a twelve sided polygon in the next row. Dragging the formulae down the table, doubling the number of sides each row, I obtained Pi to 14 decimal places in the 23rd row. (Result confirmed by Tamura and Kanada's published figures). Since Excel is limited to 15 significant figures that was the best result I could get. I consoled myself with the thought that the best mathematicians in the world had struggled for centuries to achieve that degree of accuracy.

I then repeated the exercise using the Gaussian method favoured by T and K. This achieved the same result in row 4 !