The Circumference of a Circle

Date: 11/4/95 at 10:10:46
From: dennis sartoros
Subject: Calculating the value of pi; circumference of a circle

How do scientists and mathematicians know that the circumference 
of a circle is 3.1416.. etc.?

sartoros@part.com

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Date: 11/5/95 at 15:3:35
From: Doctor Ethan
Subject: Re: Calculating the value of pi; circumference of a 
circle

Well,

I suppose that there are many ways to calculate the value of Pi.

The easiest that I know is like this.

If we understand Pi to be the value diameter/2*radius then
we can estimate this by taking polygons closer and closer to a 
circle with an diagonal length of 1/2.  

For example, a regular polygon with 30 sides looks a lot like a 
circle.

So we can estimate the value of Pi by finding the perimeter of the 
polygon.

Let's do that to see how close we get.

If we do have 30 sides then we can think of the polygon as be made 
up of 30 isosceles triangles each having up vertex angle of 360/30 
degrees.

That is 12 degrees.  So we have a 30 triangles like this

            /\
           /12\
        .5/    \
         /      \
        /84    84\
       /__________\
             A

Where 12 and 84 are angle measures and .5 and A are the lengths of 
the sides.

So to find the perimeter we just need to find the length of the 
side opposite the angle 12.

To do this we can use a handy law called the law of sines.

Which tells us that sin(84)/.5 = sin(12)/A.

A is 0.104528

so to find the perimeter we multiply by 30 to get 3.13584

Well, that is pretty close but we could get closer by taking a 
polygon of more sides.  I will just work out one here for 180 
sides to see if it gets better.  Yep, with 180 sides the perimeter 
is 3.14143, so that is starting to get really close.

I hope that this method has started to make some sense to you.  If 
not write back and we will try to help more.

-Doctor Ethan,  The Geometry Forum