Hands-on with Pi
Pi, on one hand, seems to be a simple thing – just something they teach you in school, so it can't be too complicated. But if you give Pi some thought, you will find lots of AMAZING things about it.
The definition of Pi says it is the ratio of any circle's circumference to its diameter. Students can easily find out this ratio on their own by MEASURING the circumference (with a measuring tape) and the diameter (with a ruler) of several circles, and then calculating the ratio of them. This is a great hands-on activity and a starting point for a discussion about Pi!
Students will naturally get varying results because it's impossible to measure totally accurately. They should calculate the average of the ratios they get. Hopefully at least some of them will get their average ratio as being between 3.1 and 3.2.
In reality, π is not any fraction or a simple decimal, but an irrational number – one that you cannot write as a fraction using two whole numbers. This is something quite amazinga and puzzling! Being an irrational number means that if you write π as a decimal, its decimal digits go on forever... AND without ever settling into any pattern!
Mathematicians didn't find find out about all that for thousands of years. Ancient cultures were aware of the fact that the ratio of any circle's circumference to diameter was a constant. The formulas they used for the area of circle indicate the equivalent value for π as 377/120, 3 1/8, or 256/81.
Archimedes found out that this ratio was between 3 10/71 and 3 1/7. Using decimal notation and rounding to 8 decimal places, this means Pi is between 3.14084507 and 3.142857143. The true value of Pi to 8 decimal digits is 3.141592654 so this means Archimedes got it correct to two decimal places -- and 3.14 is even today the figure given to school children to use in math book problems.
Archimedes' approach was to draw a regular polygon inside the circle and another outside it. He then calculated the perimeter of both polygons, and from that he was able to calculate between what two values Pi would lie.
Image by Leszek Krupinski
Archimedes wasn't afraid of calculations and used a polygon with 96 sides! Here's a nice interactive about Archimedes' method where you (and your students) can see how the approximation gets closer as the number of sides of the polygon increases.
Over time, mathematicians were able to calculate Pi more accurately. The proof of Pi's irrationality came in 1768 by Johann Lambert. After that, mathematicians have known for sure it is irrational – but the quest for finding more decimal digits for Pi still continues today!
Guess how many decimals they have found thus far?
As of 2013 π had been calculated to 12,100,000,000,050 decimal digits - 12.1 trillion digits!
If you'd like to start memorizing it, here's a link to Pi trainer :)