Calculating the Value of Pi

Special Properties of Circles

Circles are unique in the property of having the shortest perimeter to enclose any given area. Likewise spheres have the smallest surface area to enclose any given volume. This makes them efficient objects. They appear all throughout nature.  Planets and stars gravitationally pull themselves into spheres.  Soap bubbles form perfect spheres as a natural form to use the least amount of soap film to form their surface.  Rising air bubbles in water naturally form efficient spheres.  Rain drops falling on water form circular rings of waves moving out equally from the point where the drop hit. 

There is a relationship between the diameter of a circle and its circumference.  So if you take a 10" circle and measure its circumference you will find that it measures about 31.4 inches.  So if you divide 31.4 by 10 you will get 3.14 which is the rough value of Pi.  It doesn't matter what size the circle is; the relationship is always true.  Using Pi and the radius of any given circle you can calculate its circumance or its area.  Also with the radius of a sphere with Pi you can calculate the surface and the volume which makes knowing the exact value of Pi important.  However, there is not exact number for Pi as there appear to be an infinite number of decimal places folowing the 3.14159. . . . . This seems to reflect the nature of a circle in that you can travel around it an infinite number of times, without end. 

Finding Infinite Progressions for Pi

Going back to anicent times mathicans have sought to discover the exact value of Pi; however, there is no fraction that give the exact value for Pi.  Over the centuries a number of mathamatic progressions have been develoved which give a value to Pi, but there is no set formula which gives an exact value as all of these are progressions which go on and on without ever ending.  The decimal number for Pi does not repeat any sequence and goes for for infinity. 

The Greek mathematician Archimedes used polygons inside and outside a circle to compute some outer boundries for the value of Pi.  In his calculations he determined that Pi is greater then 223/71 (3.1408) and less that 22/7 (3.1428).  If you take the average of those two boundry numbers you get 3.1418 which is not too far off the mark, but not as good as Liu Hui below. 

In China around 265 AD mathematician Liu Hui used a 3,072 sided polygon to obtain a value of Pi = 3.1416.  


In the 1400 hundreds an Indian mathematician, Madhava discovered a infinite progression for Pi, and used the series to calculate Pi to eleven decimal places. (3.1415926535 very impressive!) In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregory in 1671, and by Leibniz in 1674. 

In this series the first term is 4, which is too great, so the second term subtracts 1 and 1/3, making Pi 2.666 which is to small.  So the next calcualtion adds 4/5, making it 3.46666, still too large. So the process of these infinite progressions is to go back and forth slowly narrowing in on the value of Pi:

If we take the mid points (shown as Red Dots) between each calcualtion we narrow down on the correct answer even faster. 



The first Pi series discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician Francois Viete in 1593:



The second Pi infinite series discovered in Europe by John Wallis in 1655, was also a product.  

These will be the infinite series that we will use in our computations.  At the bottom of the page are links to the pages for the calculations:  


With the advent of the computer, and new more powerful math programs mathematicians have been able to compute the value of Pi to extremely long decimal values. In 1949 John Wrench and Levi Smith reached 1,120 digits using a desk calculator. That same year George Reitwiesner and John von Neumann calculated Pi to 2,037 digits using an ENIAC computer.  Ten thousands digits was achieved in 1958 and 100,000 digits in 1961. In 1973 1 million digits were reached.  And we are well over 10 trillion calculated digits today (2011). Those new algorithms are able to solve 14 digits per term! But will the madness ever stop? 

With our home computers we will only be striving to reach 15 digits using the above series.  I have written Java Script programs which take the Madhava-Leibniz, Wallis, and Nilakantha infinite progression to calculate Pi. Also included are two columns of averaging to get to the answer quicker as discribed in the graph above. The last column averages the previous averaged column making it even faster.  

Things for You to check!

Click on the Links below to run the calcualtions. You will see a box which asks you to imput the number of calcualtion you want for the run. You might want to start with a modest 20 calculations and so you can study how the calcualtions start to progress.  At the bottom of the run you can click the 'Run Again' Button and do as many as you like.  It is interesting to compare the different Progression as some work quicker and more efficiently that others.  See which one you like the best.  

Madhava - Leibniz Calculations 
Wallis Calculations
Nilakantha Calculations 


June 2016