A Historical Note On The Proof Of The Area Of A Circle

Proofs that the area of a circle is ?r2 can be found in mathematical literature dating as far back as the time of the Greeks. The early proofs, e.g. Archimedes, involved dividing the circle into wedges and then fitting the wedges together in a way to approximate a rectangle. Later more sophisticated proofs relied on arguments involving infinite sequences and calculus. Generally speaking, both of these approaches are difficult to explain to unsophisticated non-mathematics majors. This paper presents a less known but interesting and intuitive proof that was introduced in the twelfth century. It discusses challenges that were made to the proof and offers simple rebuttals to those challenges.

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Some Comments On: A Historical Note On The Proof Of The Area Of A Circle

American Journal of Business Education (AJBE) ◽

10.19030/ajbe.v4i12.6613 ◽

2011 ◽

Vol 4 (12) ◽

pp. 45-50

Author(s):

Jaideep T. Naidu ◽

John F. Sanford

Keyword(s):

Twelfth Century ◽

Business School ◽

Alternative Solution ◽

Historical Note ◽

School Students ◽

Mathematics Majors ◽

Alternative Method ◽

Starting Point ◽

School Faculty ◽

Intuitive Proof

In a recent paper by Wilamowsky et al. [6], an intuitive proof of the area of the circle dating back to the twelfth century was presented. They discuss challenges made to this proof and offer simple rebuttals to these challenges. The alternative solution presented by them is simple and elegant and can be explained rather easily to non-mathematics majors. As business school faculty ourselves, we are in agreement with the authors. Our article is motivated by them and we present yet another alternative method. While we do not make an argument that our proposed method is any simpler, we do feel it may be easier to communicate to business school students. In addition, we present a solution using a rectangle which could be left as an exercise for the student after a brief explanation in class. Finding the area of a stack of rectangles with a rectangle as a starting point may seem redundant at first. However, we show that it is actually an excellent algebraic exercise for students since it offers a certain challenge which a square does not. We also solve this exercise using the quicker triangular approach and feel it can be appreciated by students in an Introduction to Calculus course. We also provide two interesting links that demonstrate the work of the ancient mathematicians for this well known problem.

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