History of Circle Area Formula

Date: 03/19/2007 at 18:26:08
From: Richard
Subject: history of equation using Pi 

Do we know who figured out that pi r squared is the area of a circle?

I can find out about the history of Pi and the circumference of a 
circle, but not its area.  I looked through your FAQs and on Google
but to no avail.  Perhaps it is just not known?

Date: 03/20/2007 at 06:08:17
From: Doctor Peterson
Subject: Re: history of equation using Pi

Hi, Richard.

It's hard to answer that question, because the area of a circle was 
known long before pi was actually used.  Proposition 2 of book XII of 
Euclid's Elements, which was undoubtedly known before Euclid 
himself, is equivalent to the formula A = pi r^2:

  Euclid's Elements Book XII, Proposition 2  
http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII2.html 

 Circles are to one another as the squares on their diameters.

That is, the area of a circle is proportional to (2r)^2, which in 
turn is proportional to r^2.  All that is lacking here is a name for 
the constant of proportionality, which has been called pi since 1706.

There are two parts to your question: who discovered that the area is
SOMETHING times the square of the radius (for which the answer is
whoever gave Euclid his proof, commonly considered to be Eudoxus); and
who discovered that the constant of proportionality is pi.  The answer
to the latter question is Archimedes.

The form in which Archimedes stated it was that the area of a circle
is equal to that of a right triangle whose base is the circumference
of the circle, and whose height is the radius of the circle.  That is,

  A = 1/2 (2 pi r) r = pi r^2

in modern terms.  So except for the lack of algebraic notation and a
name for pi, he got the entire formula.  You may be aware that he also
worked out the value of pi.

His proof can be found in Hawking's _God Created the Integers_ (a
collection of important primary documents in math history); and in
sites like the following:

   Archimedes and the Area of a Circle 
http://mathrefresher.blogspot.com/2006/05/archimedes-and-area-of-circle.html 

It is related to what I said in the following simplified explanation:

  Why Pi?
    http://mathforum.org/library/drmath/view/58292.html 


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