Attribution: This is a cached copy of a third party project. Many of these sites are from 20 years ago and the majority are no longer running. We show only the first page of the project. We do not save all pages since copyright belongs to the third-party author.
Title: The Pleasure of Pi Objectives/Goals The hypothesis of the experiment is that the ratio between the error in determining I using by inscribing polygons within and circumscribing polygons about a circle with (km) sides and that obtained using polygons with (kn) sides will approach (n/m)^2 as k increases. Methods/Materials To test my hypothesis, I needed to develop formulas to determine the perimeters of the regular polygons inscribed within and circumscribed about a circle. I discovered that the perimeter of the regular polygon with X sides inscribed in a circle with a diameter of 1 is X(sin(180/X)). The perimeter of the regular polygon of X sides circumscribed about a circle with a diameter of 1 is X(tan(180/X). I estimated I by using the expression: (X(sin(180/X) + X(tan(180/X))) / 2, and I calculated the error in estimating pi using polygons with the formula: error = ((X(sin(180/X) + X(tan(180/X))) / 2) - I. I calculated the ratios of the errors of the estimates using polygons of m and n sides employing six different values for m and n [(m=8, n=10,) (m=6, n=8,) (m=4, n=6,) (m=4, n=8,) (m=4, n=10,) and (m=4, n=12)]. I then calculated the error ratios for polygons of km and kn sides using those given m and n values, and k = (1, 2, 3, 4, and 1000). Finally, I graphed the results. Results The graphs are consistent with the hypothesis. As k increases, the error ratio approaches (n/m)^2, the square of the inverse of the ratio of the number of sides. Conclusions/Discussion By completing this experiment, I discovered that the ratio between the error in determining I using by inscribing polygons within and circumscribing polygons about a circle with (km) and (kn) sides approaches (n/m)^2 as k increases. Summary Statement The summary is that I determined that the ratio between the error in estimating pi using by inscribing polygons within and circumscribing polygons about a circle with (km) and (kn) sides approaches (n/m)^2 as k increases. Help Received Dad helped edit my report.