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Deriving Pi

Date: 05/10/2001 at 22:02:57
From: Leslie Hannah
Subject: How is Pi actually derived from the beginning to the end?

I know that Pi is equivalent to 3.14 but what is the formula used to 
come up with the 3.14? I am aware of the way you use the formulas for 
a circle but could you explain to me each step of the process of 
deriving the formula to get 3.14? Thanks.

Date: 05/11/2001 at 16:38:48
From: Doctor Rob
Subject: Re: How is Pi actually derived from the beginning to the end?

Thanks for writing to Ask Dr. Math, Leslie.

The way that Archimedes and others up to the end of the Middle Ages
used to compute Pi was to approximate it using a regular polygon of
n sides and its inscribed and circumscribed circles.  The inscribed
circle has circumference smaller than the perimeter of the polygon,
which is in turn smaller than the circumference of the circumscribed
circle.  That gave inequalities of the form

   P/(2*r) > Pi > P/(2*R)

By using very large values of n, the first and last of these can be
made very close together, which gives a very good estimate of Pi.

These inequalities can be rewritten in terms of n, the number of 
sides,using trigonometric functions, as

   n*tan(180/n degrees) > Pi > n*sin(180/n degrees)

Archimedes started with a regular hexagon, n = 6.  Then 180/6 = 30
degrees is the pertinent angle, and this gives

   tan(30 degrees) = 1/sqrt(3),
   sin(30 degrees) = 1/2.

This produces the inequalities

   2*sqrt(3) > Pi > 3
   3.464 > Pi > 3

If you double the number of sides to 12, you will cut the angle in 
half. You can find the tangent and sine of 15 degrees by using the 

   tan(x/2) = (sqrt[1+tan^2(x)]-1)/tan(x)
   sin(x/2) = sqrt[(1-sqrt[1-sin^2(x)])/2]

That will give you the values

   tan(15 degrees) = 2 - sqrt(3) = 0.267949...
   sin(15 degrees) = sqrt[2-sqrt(3)]/2 = 0.258819...


   3.21539 > Pi > 3.105829

Doubling the number of sides to 24, you get

   tan(7.5 degrees) = 0.13165250
   sin(7.5 degrees) = 0.13052619
   3.15966 > Pi > 3.13263

Doubling again to 48 sides, you get

   tan(3.75 degrees) = 0.06540313
   sin(3.75 degrees) = 0.06554346
   3.14609 > Pi > 3.13935

Doubling again to 96 sides, you get

   tan(1.875 degrees) = 0.03273661
   sin(1.875 degrees) = 0.03271908
   3.14271 > Pi > 3.14103

This already shows that the first three significant figures of Pi are
3.14. This can be continued to get more and more significant figures
of Pi. Ludolph Van Ceulen used this method to compute 17 decimal
places of Pi in the early 1600s, which was a record at the time.  
To 20 decimal places, you get

   Pi = 3.14159265358979323846...

Modern methods of computing Pi are somewhat different. This is a very
complicated and interesting subject, about which I can't go into much
more detail here.

For more, see the following Web page from the Dr. Math Frequently 
Asked Questions (FAQ):

  About Pi   

- Doctor Rob, The Math Forum   
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School History/Biography
Middle School Geometry
Middle School History/Biography
Middle School Pi
Middle School Two-Dimensional Geometry

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