Re: I seem to have forgotten...
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Michael Grodsky writes:
> I seem to have forgotten some basic math...would someone
> please rehash, remind, reteach to me why the area of a circle
> is pi*r^2, why the circumfrence of a circle is 2pi*r and why
> and how pi (the omnipotent, omniscient, and god-like) was
> figured to be the ratio of a circle's diameter to it's
> circumfrence.
The questions raised here are by no means elementary,
though they may be "basic". There is no _a priori_ reason
why the number pi _should_ appear in the calculation of the
area or the circumference of a circle, or that these two
measurements should be related at all. The gradual, growing
realization of the significance of pi and of the inscrutable
nature of this number are part of the history of civilization
as we know it. This story is best told (to my knowledge) in
the book "A History of Pi", by Petr Beckmann (3rd ed. 1971,
St. Martin's Press). A very readable book, with a million
fascinating sidelights.
To get started on the "Why?", you can imagine a circle of
radius 1 covered by four unit squares, so that its area must be
less than four. If you cut off half of each square, diagonally,
you will see that the circle _covers_ a square with area 4(1/2)
= 2, so that the area of a circle is greater than 2. And so on..
Along the way you might run into Kurschak's (accents ommitted)
tile, which is a way of showing that the dodecagon (12 sides)
inscribed in a unit circle has area exactly 3, so that the area
of the circle is just a little over 3. For a reference to this,
see "The Penguin Dictionary of Curious and Interesting Geometry"
by David Wells (Penguin, 1991).
It is always refreshing to have someone ask "why" rather
than "how"! Don't worry about just starting out -- we are all
on the road somewhere...
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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