Amazing Things To Note About Circles and Area (original) (raw)
International Journal of Science and Mathematics …, 2010
The issue of the area of irregular shapes is absent from the modern mathematical textbooks in elementary education in Greece. However, there exists a collection of books written for educational purposes by famous Greek scholars dating from the eighteenth century, which propose certain techniques concerning the estimation of the area of such shapes. We claim that, when students deal for an adequate period with a succession of carefully designed tasks of the same conceptual basis-in our case that of the area of irregular shapes-then they "reinvent" problem-solving techniques for the estimation of their area, given that they have not been taught anything about these shapes. These techniques, in some cases, are almost the same as the abovementioned historical ones. In other cases, they could be considered to be an adaptation or extension of these.
Measurement and Decomposition: Making Sense of the Area of a Circle
International Group for the Psychology of Mathematics Education, 2019
As a component of a course on geometry for preservice elementary teachers (PSTs), we derive area formulas for a variety of polygons including triangles, quadrilaterals, and both regular and irregular shapes whose areas can be measured empirically using decomposition. Decomposing a circle to justify why its area can be measured using the standard formula is more challenging as it requires both empirical and deductive reasoning involving limits. In spite of the challenge, we expected decomposition strategies to transcend work with polygons and support PSTs when thinking about the area of circles. Results show that few PSTs utilized decomposition and instead focused on finding meaning in the symbolism of the formula. Concept images related to area will be discussed.
Journal of Automated Reasoning
The area method for Euclidean constructive geometry was proposed by Chou, Gao and Zhang in the early 1990’s. The method can efficiently prove many non-trivial geometry theorems and is one of the most interesting and most successful methods for automated theorem proving in geometry. The method produces proofs that are often very concise and human-readable. In this paper, we provide a first complete presentation of the method. We provide both algorithmic and implementation details that were omitted in the original presentations. We also give a variant of Chou, Gao and Zhang’s axiom system. Based on this axiom system, we proved formally all the lemmas needed by the method and its soundness using the Coq proof assistant. To our knowledge, apart from the original implementation by the authors who first proposed the method, there are only three implementations more. Although the basic idea of the method is simple, implementing it is a very challenging task because of a number of details that has to be dealt with. With the description of the method given in this paper, implementing the method should be still complex, but a straightforward task. In the paper we describe all these implementations and also some of their applications.
Initially, this work deals with the fundamentally significant concept of geometric transformation which is responsible for the classification of Geometries. Afterwards, the Euclidean transformations such as translation, axial symmetry and rotation are presented as special cases of isometries, since they conserve the distances and the measures of angles. Data research, contrary to the cognitive regards, shows that medium school age students are confused in the implementation of such transformations. This study proposes that Euclidean transformations can be comprehended, via the exploitation of the tools of Dynamic Geometry System of Cabri Geometry II. Then these transformations are considered as cognitive vehicles so that they may become helpers in assessing the area of basic shapes of plane geometry such as parallelograms, triangles, trapeziums, circles and polygons and proving simple identities through the method of splitting parts of an object and recomposing them. Finally, anticipating the benefits of consolidation, the Haberdasher's Puzzle is proven by cutting an equilateral triangle into four pieces that may be rearranged in order to form a square through Euclidean transformations.
Area Enclosed by Tangential Circles
International Journal of Scientific Research in Computer Science, Engineering and Information Technology, 2020
The problem in this paper finds its roots in trigonometry and recreational pure mathematics. Given n tangential circles having radius r, the problem asks to find the area enclosed by these circles. The paper takes a logical, step-by-step approach to break down the problem into several sub-parts, and then use those to derive the final solution.
The Area Method - A Recapitulation
J. Autom. Reason., 2012
The area method for Euclidean constructive geometry was pro posed by Chou, Gao and Zhang in the early 1990’s. The method can efficiently p rove many non-trivial geometry theorems and is one of the most interesting and most su cce sful methods for automated theorem proving in geometry. The method produces proo fs that are often very concise and human-readable. In this paper, we provide a first complete presentation of the method. We provide both algorithmic and implementation details that were omitted i n the original presentations. We also give a variant of Chou, Gao and Zhang’s axiom system. Bas ed on this axiom system, we proved formally all the lemmas needed by the method and its soundness using the Coq proof assistant. To our knowledge, apart from the original implementation by the authors who first proposed the method, there are only three implementations more . Although the basic idea of the method is simple, implementing it is a very challenging task because of a number of de...
On the Exact Measurement And Quadrature of The Circle Second Edition
On the Exact Measurement And Quadrature of The Circle Second Edition
In the spring of 1985, while helping the son of an acquaintance prepare for a high school examination in Greek geometry, I experienced an overwhelming personal and compelling insight into the nature of the constant now called pi. From two very careful experimental tests of my hypothesis, I found that the ratio cited universally as the definitive numerical value of pi, does not accurately quantify the circumference of a circle, but that pi is measurably greater than 3.14159…. Over months of personal research, I gradually focused my studies upon the nature of circular motion as described by the great Galileo Galilei, and I eventually developed a logical and mathematical argument by which I decomposed circular motion into two simultaneous rectilinear component movements, whose separate magnitudes, and whose compounded magnitude, could be determined exactly by means of Greek geometry.
Some Comments On: A Historical Note On The Proof Of The Area Of A Circle
American Journal of Business Education (AJBE)
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...
A Mechanical Derivation of the Area of the Sphere
The American Mathematical Monthly, 2001
In the beginning of the twelfth century CE, an interesting new geometry book appeared: The Book of Mensuration of the Earth and its Division, by Rabbi Abraham Bar Hiya (acronym RABH), a Jewish philosopher and scientist. This book is interesting both historically and mathematically. Its historical aspect is discussed in [3]. In this paper we consider the mathematical aspect. The second part of the book contains a beautiful mechanical derivation of the area of the disk [5, §95]. Roughly speaking, the argument goes as follows (see Figure 1): The disk is viewed as the collection of all the concentric circles it contains. If we cut
The proof of Twin ratio for finding area of circles by using chords
From the ancient civilizations of Egyptians pharaohs andGreeks, Theyknew the shape of circle and used it on their temples and tombs. They described circles by using diameters and after that, manytrials have done to find the relation between area of circles and diameters by using π, which results from the division area of circle over its radius square. Unfortunately, All chords have been neglected because there is no specific description for circles by using chordsas well as there is no a real explanation for the ratio result from division area of circle over square of any chords because it change from chord to another. In this paper, we managed to make a new description for circles by chords (golden description)in addition to proving a ratio can deal with all chords and diameters to determine the area of circle and called it (twin ratio).
Computation of Circular Area and Spherical Volume Invariants via Boundary Integrals
SIAM Journal on Imaging Sciences, 2020
We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our results also extend to higher dimensional hypersurfaces. The resulting surface integrals are computable analytically on a triangulated mesh. This gives a simple computational algorithm for computing the spherical volume invariant for triangulated surfaces that does not involve discretizing the ambient space. We discuss potential applications to feature detection on broken bone fragments of interest in anthropology.
A Historical Note On The Proof Of The Area Of A Circle
Journal of College Teaching & Learning (TLC), 2011
Proofs that the area of a circle is πr 2 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.
A Method for the Squaring of a Circle
Advances in Pure Mathematics
This paper presents a Method for the squaring of a circle (i.e., constructing a square having an area equal to that of a given circle). The construction, when applied to a given circle having an area of 12.7 cm 2 , it produced a square having an area of 12.7 cm 2 , using only an unmarked ruler and a compass. This result was a clear demonstration that not only is the construction valid for the squaring of a circle but also for achieving absolute results (independent of the number pi (π) and in a finite number of steps) when carried out with precision.
Review - Paul Calter, Squaring the Circle.pdf
Any one who has googled the words "art" and "mathematics" in the same phrase has chanced upon Paul Calter's website on art and geometry. Crafted for his course at Dartmouth University, these pages offer generous examples of art through history that evinces the mathematical knowledge of its day.
THE AREA INDUCED BY CIRCLES OF PARTITION AND APPLICATIONS
THE AREA INDUCED BY CIRCLES OF PARTITION AND APPLICATIONS, 2022
In this paper we continue with the development of the circles of partitions by introducing the notion of the area induced by circles of partitions and explore some applications.
THE AREA METHOD AND APPLICATIONS
THE AREA METHOD AND APPLICATIONS, 2021
In this paper we develop a general method for estimating correlations of the forms \begin{align}\sum \limits_{n\leq x}G(n)G(x-n),\nonumber \end{align}and \begin{align}\sum \limits_{n\leq x}G(n)G(n+l)\nonumber \end{align}for a fixed 1leqlleqx1\leq l\leq x1leqlleqx and where G:mathbbNlongrightarrowmathbbR+G:\mathbb{N}\longrightarrow \mathbb{R}^{+}G:mathbbNlongrightarrowmathbbR+. To distinguish between the two types of correlations, we call the first \textbf{type} 222 correlation and the second \textbf{type} 111 correlation. As an application we estimate the lower bound for the \textbf{type} 222 correlation of the master function given by \begin{align}\sum \limits_{n\leq x}\Upsilon(n)\Upsilon(n+l_0)\geq (1+o(1))\frac{x}{2\mathcal{C}(l_0)}\log \log ^2x,\nonumber \end{align}provided Upsilon(n)Upsilon(n+l_0)>0\Upsilon(n)\Upsilon(n+l_0)>0Upsilon(n)Upsilon(n+l_0)>0. We also use this method to provide a first proof of the twin prime conjecture by showing that \begin{align}\sum \limits_{n\leq x}\Lambda(n)\Lambda(n+2)\geq (1+o(1))\frac{x}{2\mathcal{C}(2)}\nonumber \end{align}for some mathcalC:=mathcalC(2)>0\mathcal{C}:=\mathcal{C}(2)>0mathcalC:=mathcalC(2)>0.