Why Pi?

Date: 01/25/2001 at 11:41:06
From: crystal
Subject: Why Pi?

Dr. Math, I was just wondering... 

Why do we use pi when we calculate the circumference and area of a 
circle? I think one of my professors once told my class but I can't 
remember and am curious.
 
Thanks.

Date: 01/25/2001 at 13:06:19
From: Doctor Peterson
Subject: Re: Why Pi?

Hi, Crystal.

There are two questions here, with very different answers.

First, for the circumference, it's because we DEFINE pi as C/D, so we 
can write C = pi D automatically. There's a trick hidden behind that 
definition, though: how do we know that C/D is the same for every 
circle? That takes a bit of proof, and leads to some interesting 
ideas; look in the Dr. Math FAQ on pi:

  Pi = 3.14159...
  http://mathforum.org/dr.math/faq/faq.pi.html   

or in the following answers in particular, for an explanation:

  Why is Pi a Constant?
  http://mathforum.org/library/drmath/view/57828.html   

  Einstein, Curved Space, and Pi
  http://mathforum.org/library/drmath/view/55198.html   

  Is Pi a Constant in Non-Euclidean Geometry?
  http://mathforum.org/library/drmath/view/55021.html   


For the area, there's a nice way to see why the formula should be what 
it is.

Let's think about regular polygons first, and look at the relation 
between their areas and perimeters. Any n-sided polygon can be broken 
into n isosceles triangles like this:

        +-----+
       / \   / \
      /   \ /   \
     +-----+-----+      +
      \   / \   /      /|\
       \ /   \ /      / |a\
        +-----+      +--+--+
                        s

Each of these triangles has a base that is equal to a side s of the 
polygon, and a height a (called the apothem); the total area is

    A = n * sa/2

This can be rearranged as

    A = (ns)a/2

and since ns is just the perimeter P of the polygon, this means

    A = Pa/2

Now make n very large, and a will be very close to the radius r of the 
circle the polygon is becoming. We can see (and could prove more 
carefully if we took the time) that for a circle,

    A = Cr/2

where C is the circumference (perimeter) and r is the radius.

But since we know

    C = 2 pi r

this becomes

    A = 2 pi r * r/2 = pi r^2

We're done! Because we could find the area of a polygon using its 
perimeter, we can find the area of a circle using its circumference, 
and that uses pi.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/