This project was done in order to see how accurately PI could be calculated by hand. By the way, "calculated by hand" means: no calculater, no slide rule, no log / trig table, no ruler, and also no trigonometry or anything else that is derived from knowing what PI is). Everything (including the basic arithmetic calculations) were done by hand. I did use a ruler; however, it was one that I crafted out of a piece of wood, my fingernail, and a knife. The original calculations were done without graph paper; the graphics below were reproductions that were done on graph paper at a larger scale for the purpose of viewing on the web. Read the rest of the article to find out how it was done!
The above diagram is the one that I used for all of my geometric calculations. It represents one quarter of an octogon with a radius of six; a figure with a larger number of sides could have been used for more precision. At this point, I will not explain what the various letters mean. Rather, I will explain them as they are used, and let you refer back to the diagram at your convenience.
I. Calculate the Circumfrence with the Pythagorean Theorum
The first thing that must be done is to calculate the circumfrence (8*[d]) of our octogon. This is accomplished by the use of a small amount of algebra (although it wasn't necessary if you simply measured the distance with a constructed ruler). Anyway, because this is an octogon, we know that the slope of [L2] is m=1 (m=^y/^x | m=6/6 | m=1). Also, we know that the y-intercept of [L2] is (0,6). With this knowledge, we can calculate the equation of [L2], which we find to be (y = x + 6).
After finding that, we can proceed to calculate the length of segment [a]. This is done by using the formula to calculate the distance between a point and a line, where the line is [L2] and the point is the origin (see the diagram below for how this is done). If the equation is used, a square root is usually encountered; see below for how to arithmetically solve this.
Furthermore, we know that ([a] + [b] = 6), because that is a radius of the octogon, so we can calulate [b] by subtracting [a] from six. It is even easier to calculate [e]: just use the distance formula between (0,6) and (-6,0) and divide the result by two. Finally, use the pythagorean theorum to calculate [d], and multiply it by (8) to get the circumfrence ([c]).
II. Create an Equation by Solving for PI
Now that we have the circumfrence, we can solve for PI using the formula for the circumfrence of a circle. With our values from above, we find that PI is (36.8 / 12).
III. Compute a Value for PI
Now all that is left to do is to use some simple long division to calculate the value of PI. In the end, I arrived at a value of 3.1333333333, which is reasonable accurate considering that only an octogon was used. If you wanted more precision, you could just use a more complex regular polygon (I have done this, but the calculations for an octogon end up working in a way that is very nice and easy to understand for a web article).
IV. Who am I, and Why Did I Write This?
My name is Jon Hardin and I am a student in Madison, WI. I write software for Windows and Linux in my spare time, as well as work on math projects such as this one. I run Intelli-Computing Software, which has published several PC-based computer games, but aside from that I enjoy solving interesting math problems and developing new methods of deriving old equations and constants. As a result, I have prepared this article to describe the derivation of one of my favorite numbers, PI.
