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/
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