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Tom Richmond

Western Kentucky University
1 Big Red Way
Bowling Green, KY 42101

 
tom.richmond@wku.edu
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Derivation of the Formula for Area of a Circle


Deposit # 85


Another approach:
NCB Deposit  # 17


 

 
While firmly based in Western Civilization, this animation uses almost exactly the same approach in deriving the formula for the area of a circle as that found in the oldest book from China,  The Nine Chapters on the Mathematical Art, by Liu Hui (ca. 250 AD). . . .

Chapter One, Problem #32.  Chinese No. 32

         A circular field has a perimeter of 181 steps and a diameter of 60 and 1/3 steps?  What is its area?

         Answer:  The area equals half the perimeter times half the diameter.
Formula
From the "Nine Chapters"



Recall the study of cones in Western Civilization dates to Apollnius (262-190 BC) and other early Greeks.
Four conic sections
The circle, ellipse, parabola and hyperbola are "sections" of  a cone.
Hyperbola

References

For a sampler of Richmond's other work, see < http://www.wku.edu/~tom.richmond  >.

Tom Richmond and Aaron Young, Instant Insanity II, College Mathematics Journal, Vol. 44, no. 4 (Sept. 2013) 265-272.

For the Nine Chapters see < http://curvebank.calstatela.edu/popdowns/th/th26/th26.htm >
and in Chinese, see < http://www.chinapage.com/math/s9/s9.html  >.
For a translation, see Shen Kangshen, John N. Crossley and Anthony W.-C Lun, The Nine Chapters on the Mathematical Art, Oxford, 1999.
Also in translation, see Man-Keung SIU, "An Excursion in Ancient Chinese Mathematics," in Using History to Teach Mathematics:  An International Perspective, ed. V. Katz, MAA, 2000, p. 163.

For a MATHEMATICA® reference, see
< http://mathworld.wolfram.com/Circle.html  >.

For a Maple® source, open < http://www.mapleapps.com  > and type "circle" or "conic" in the search box.

Beckmann, Petr,  A History of    . St. Martin's Press, 1971, p. 18.

Jennings, George A., Modern Geometry with Applications, Springer-Verlag,  1994, p. 20.

Stewart, James, Calculus,  5th ed., Thomson: Brooks/Cole, 2003, pp. A16 and 527.


and on history in Mathematics . . .

Before leaving the above images, we invite the viewer to consider the following:  In early Propositions in the Elements, Euclid investigated the now classic shapes found in all geometry books.  To see the first page of the first printed edition of the Elements click here.

For more on the Nine Chapters in the Shanghai Library as pictured on the right,
<
http://curvebank.calstatela.edu/popdowns/th/th26/th26.htm >.


For Lui Hui, see
<
http://curvebank.calstatela.edu/popdowns/th/th11/th11.htm >.
Main library in Shanghai
Bettina and Tom Richmond have collaborated on several projects, including an earlier animation for the NCB.  She is an algebraist and he is a topologist.

Signature
The NCB also thanks Dr. Derek Chang, CSULA, Dr. Man-Keung SIU, University of Hong Kong and Zhou Yugin, Shanghai Library.
Chang signature
Siu signature

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