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### 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/
```
Associated Topics:
Middle School Pi

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