The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

History of Calculating Pi

Date: 05/20/98 at 15:57:40
From: Josh
Subject: What is the formula for pi?


I was doing a report on pi and wondered what the formula for pi was. 
How did they figure out that it is:
   3.14159265358979323846 ...


Date: 05/20/98 at 19:12:01
From: Doctor Wilkinson
Subject: Re: What is the formula for pi?

This is a very good question.

The first mathematician to calculate pi with reasonable accuracy was
Archimedes, around 250 B.C. Using the formula:

   A = pi r^2

for the area of a circle, he approximated pi by considering regular 
polygons with many sides inscribed in and circumscribed around a 
circle. Since the area of the circle is between the areas of the 
inscribed and circumscribed polygons, you can use the areas of the 
polygons (which can be computed just using the Pythagorean Theorem) to 
get upper and lower bounds for the area of the circle. This was the 
first general method for calculating approximations to pi, and at 
least theoretically it could be used to get any degree of accuracy if 
you could just do the computations. Archimedes showed in this way that 
pi is between 3 1/7 and 3 10/71. The same method was used by the early 
seventeenth century with polygons with more and more sides to compute 
pi to 35 decimal places (Van Ceulen did the biggest 674 calculations.)

When Newton and Leibnitz developed calculus in the late seventeenth 
century, more formulas were discovered that could be used to compute 
pi. For example, there is a formula for the arctangent function:

   arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...

If you substitute x = 1 and notice that arctan(1) is pi/4 you get a 
formula for pi. This is not useful because it takes too many terms to 
get any accuracy, but there are some related formulas that are very 
useful. The most famous of this is Machin's formula:

   pi/4 = 4 arctan(1/5) - arctan(1/239)

This formula and similar ones were used to push the accuracy of
approximations to pi to over 500 decimal places by the early 
eighteenth century (this was all hand calculation!)

In the twentieth century there have been two important developments:  
the invention of electronic computers and the discovery of much more 
powerful formulas for pi. For example, in 1910 the great Indian 
mathematician Ramanujan discovered a formula that in 1985 was used to 
compute pi to 17 million digits.

Other even better methods have been developed since, and computers are
getting ever more powerful. The current record is about 51 billion 
decimal places.

-Doctor Wilkinson,  The Math Forum
Check out our web site!   

Date: 04/05/2007 at 15:21:41
From: Shantea
Subject: first computer calculation of pi?

Who came up with the first computer calculation of pi?

Date: 04/05/2007 at 22:07:15
From: Doctor Wilkinson
Subject: Re: first computer calculation of pi?

That's a very good question, Shantea!  It seems to have been George
Reitwiesner who first used an electronic computer to calculate pi, 
in 1949 on an ENIAC, a very early computer.  He computed 2037 decimal 
places.  See 

  A Chronology of Pi

for more information.

- Doctor Wilkinson, The Math Forum
Check out our web site! 

Date: 02/28/2003 at 12:58:36
From: Ahmed
Subject: Proving Machin's formula

Machin's formula says :
pi/4= 4arctan(1/5)- arctan(1/239)
could you please prove this formula ?

What is the relation between the numbers (pi/4), (1/5) , (1/239) ?

Thanks in advance.

Date: 02/28/2003 at 18:13:18
From: Doctor Wilkinson
Subject: Re: Proving Machin's formula

Hi, Ahmed.  

There's a simple proof at

   Trigonometric Equivalences - McRae

- Doctor Wilkinson, The Math Forum
Check out our web site! 
Associated Topics:
High School History/Biography
Middle School History/Biography
Middle School Pi

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.