Amazing Things To Note About Circles and Area (original) (raw)
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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.