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New Insight into Cycloidal Areas

by Tom M. Apostol (California Institute of Technology) and Mamikon A. Mnatsakanian (California Institute of Technology)

Award: Lester R. Ford

Year of Award: 2010

Publication Information: The American Mathematical Monthly, vol. 116, no. 7, August-September 2009, pp. 598-611.

Summary: The authors open the article by generalizing the fact that the area under a cycloidal arch is three times the area of the generating circle. Their first result states that this area relationship holds throughout the generation of the cycloid. That is, the area of the cycloidal sector at each instant of its generation is three times the area of the circular segment determined by the portion of the perimeter through which the circle has rolled. This is a simple application of Mamikon's sweeping-tangent theorem: "The area swept out by a collection of tangent vectors to a curve is preserved if the tangent vectors are parallelly translated to share a basepoint" (see Apostol-Mnatsakanian, American Mathematical Monthly, 109 (2002) 900-908. Various generalizations to epi- and hypo-cycloids are derived as are other cycloidal quadratures.

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About the Authors: (From the Prizes and Awards booklet, MathFest 2010) 

Tom M. Apostol joined the Caltech faculty in 1950 and is now Professor Emeritus. He is internationally known for his textbooks on calculus, analysis, and analytic number theory, (translated into six languages), and for creating Project MATHEMATICS!, a video series that brings mathematics to life with computer animation, live action, music, and special effects. The videos won first-place honors at a dozen international festivals, and were translated into Hebrew, Portuguese, French, and Spanish. He has published 102 research papers, has written two chapters for the Digital Library of Mathematical Functions (2010), and is coauthor of three texts for the physics telecourse: The Mechanical Universe…and Beyond.

He has received several awards for research and teaching. In 1978 he was a visiting professor at the University of Patras, Greece, and in 2001 was elected a Corresponding Member of the Academy of Athens, where he delivered his inaugural lecture in Greek.

Mamikon Mnatsakanian is a former Soviet doctor of sciences in theoretical and mathematical physics and astrophysics, and professor at Yerevan State University in Armenia. He is the author of a hundred published scientific and popular articles. Mnatsakanian developed 'Generalized General Theory of Relativity with Variable Gravitational Constant‘ and 'New Apparatus of Radiation Transfer Theory‘, and has created hundreds of educational games and puzzles (

As a student he invented 'CaKuLuS'-- a simple, visual and dynamic approach for solving advanced calculus problems without formulas or equations. He has taught it successfully to students of various levels in the United States. After the devastating earthquake in Armenia in 1988, he began seismic safety investigations in cooperation with Californian specialists. As the Soviet Union collapsed, he stayed in the USA where he found a true appreciation of his new friends and colleagues. In 1996 he met his dream colleague, Professor Tom Apostol, and works with him at Caltech‘s Project MATHEMATICS!


Subject classification(s): Geometry and Topology | Plane Geometry
Publication Date: 
Wednesday, August 11, 2010