A detailed description of the derivation of pi using Archimedes’ method would not fit into the scope of this paper. It’s an important idea but it would be getting too far from the main topic, that of looking at global change through high school basic geometry.
Early mathematicians had a real problem when it came to measuring things which were circular. The tools with which they worked were linear by nature. For most this was a serious stumbling block. About 300 to 200 B.C. while examining the basic parts of the circle, a circle’s diameter and its circumference, Archimedes thought that there must be some sort of a constant which bound them. He experimented with the idea that maybe if you divided the circumference of a circle by its diameter you would find a constant value. Today we call this constant pi (3. 14159....).
As in real life, if you have difficulty trying to do a particular thing, do you stop and give up? No! You draw on your past experiences.! Archimedes knew how to measure the perimeter of a regular polygon and he knew that when you increase the sides of a regular polygon from 3 to infinity, it goes from being an equilateral triangle to a circle. Archimedes now had the key to his problem.
His approach was simple. He would take a circle and do two things to it. First he constructed a regular polygon of 96 sides around and tangent to it. This polygon’s perimeter would be a bit larger than the circle. Next he constructed another regular 96 sided polygon tangent to but inside of this same circle. True this polygon’s perimeter would a little less than the circle’s perimeter, but in the words of Archimedes “not to worry”. All three figures enjoyed the same “diameter”. Dover books does a fine job detailing Archimede’s calculation of pi, read it.
When you figure the ratio of the major polygon perimeter to its diameter and then the ratio of the minor polygon perimeter to it diameter it would follow that the circle’s ratio must be some where in between. And so it was. Pi as this new number was called showed up as being between 3.14103 > 3.141>3.14271. Today we tend to use pi as 3.14 as our classroom standard, unless we have a calculator which will give us 3.141592654.
Figure available in printed form
In observing a child at play in the dirt or in the sand at a beach we notice that he or she will make constant swirls and circles. This act has been going on for thousands and thousands of years and has lead to some basic questions from inquisitive students. What is the measure of the distance across the circle (diameter)? How do I find the measure of the circumference? And lastly how do we determine the number of square units this circle contains (area)?
The first question can be answered by most school children with a hands on demonstration. First a circle must be drawn. Now pick a random point on the circumference and anchor a string to it at that point. Next keeping the string taught swing an arc over the circle and observe how the points on the circumference “move along” the string to a maximum length and then begin to recede. The diameter’s end will be at the maximum length.
Figure available in printed form