Abstract
Intuitively, the area of a plane region should be a positive number that we associate to the region and that serves to quantify the space it occupies. We refer the interested reader to the excellent book of E. Moise for a proof that it is indeed possible to associate to each convex polygon in the plane a notion of area satisfying postulates 1. to 5. below. Our purpose in this chapter is mainly to operationalize the computation of areas, extracting from it some interesting applications.
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Notes
- 1.
Although we have not formally defined a notion of congruence for general convex polygons, the idea is pretty much the same as that for triangles: that one polygon can be moved in space, without being deformed, until it coincides with the other. In particular, note that two squares of equal sides are congruent.
- 2.
A little geometric intuition shows that this item is not independent from the others. In fact, one can prove that, under such a situation, the larger polygon can be partitioned into a finite number of other convex polygons, one of which is the smaller one. Once this has been done, a simple application of items 1. and 3. let us derive 4. as a theorem.
- 3.
Farkas Bolyai, Hungarian mathematician, and Paul Gerwien , amateur German mathematician, both of the nineteenth century.
- 4.
Alexis Claude Clairaut , French astronomer and mathematician of the eighteenth century.
- 5.
Paul Erdös , Hungarian mathematician, and Louis Mordell , British mathematician, both of the twentieth century.
- 6.
Lazare Carnot , French mathematician of the eighteenth and nineteenth centuries, the first to systematically use oriented line segments in Geometry.
- 7.
Émile Lemoine , French mathematician of the nineteenth century.
- 8.
- 9.
Such portions of Γ1 and Γ2 are usually referred to as the lunes of Hippocrates , in honor of the Greek astronomer and mathematician of the fourth century bc Hippocrates of Chios .
References
V.G. Boltianski, Figuras Equivalentes y Equicompuestas. Lecciones Populares de Matemáticas. (In Spanish.) (MIR, Moscow, 1981)
A. Caminha, An Excursion Through Elementary Mathematics I - Real Numbers and Functions (Springer, New York, 2017)
A. Caminha, An Excursion Through Elementary Mathematics III - Discrete Mathematics and Polynomial Algebra (Springer, New York, 2018)
D.G. de Figueiredo, Números Irracionais e Transcendentes. (In Portuguese.) (SBM, Rio de Janeiro, 2002)
S. Lang, Algebra (Springer, New York, 2002)
E.E. Moise, Elementary Geometry from an Advanced Standpoint (Addison-Wesley, Boston, 1963)
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Caminha Muniz Neto, A. (2018). Area of Plane Figures. In: An Excursion through Elementary Mathematics, Volume II. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77974-4_5
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DOI: https://doi.org/10.1007/978-3-319-77974-4_5
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