Scholars Journal of Physics, Mathematics and Statistics

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.

CIRCLE GEOMETRY SIMPLIFIED For Senior Secondary and Preliminary University students , 2021

My first motivation for writing this book came directly from a personal struggle faced during my secondary school days. The concept of circle geometry was completely a nightmare to me as I watched my teacher drew several circles and intersecting lines, within and without. Secondary, my teaching experience of over four years has taught me that several senior school students face similar challenges in the topic as I did during my days. I write this book to address these areas that pose a challenge to many a student who have eventually developed a phobia of the topic. Importantly also, this book is written to prepare final year students who will write external examinations such as the West African Examination Council (WAEC), National Examination Council (NECO), Unified Tertiary Matriculation Examination (UTME) and Post UTME as circle geometry happens to be one of the subjects that dominate these examinations. Preliminary and first year students at the University level can also find this book useful as it addresses some fundamental components of the curriculum. Potential engineering students can lay hold on this material at their preliminary stage, as it prepares them for important engineering components like technical drawing. To deal with this topic exhaustively, this book has been segmented into six chapters covering ten (10) fundamental theorems. Chapter one introduces circle geometry, chapter two treats theorems on chords, chapter three deals with segment theorems. Chapter four deals with cyclic quadrilateral theorems while chapter five handles tangent theorems. Chapter six treats a variety of questions cutting across all the ten theorems covered in this book. These questions and their solutions are purposely targeted at WAEC, NECO and UTME as majority of them are pulled from past questions of these examinations. The exercises at the end are meant to challenge students to familiarize themselves with the application of the theorems treated earlier. Answers are also provided for students to confirm their solutions.

International Journal of Mathematical Education in Science and Technology, 2017

The main purpose of the paper is to present a new proof of the two celebrated theorems: one is "Ptolemy's Theorem" which explains the relation between the sides and diagonals of a cyclic quadrilateral and another is "Nine Point Circle Theorem" which states that in any arbitrary triangle the three midpoints of the sides, the three feet of altitudes, the three midpoints of line segments formed by joining the vertices and Orthocenter, total nine points are concyclic. Our new proof is based on a metric relation of circumcenter.

This paper discusses some outside the box concepts with regards to circles and area computation

2010

Given $\Delta ABC$ and angles $\alpha,\beta,\gamma\in(0,\pi)$ with $\alpha+\beta+\gamma=\pi$, we study the properties of the triangle $DEF$ which satisfies: (i) $D\in BC$, $E\in AC$, $F\in AB$, (ii) $\aangle D=\alpha$, $\aangle E=\beta$, $\aangle F=\gamma$, (iii) $\Delta DEF$ has the minimal area in the class of triangles satisfying (i) and (ii). In particular, we show that minimizer $\Delta DEF$, exists, is unique and is a pedal triangle, corresponding to a certain pedal point $P$. Permuting the roles played by the angles $\alpha,\beta,\gamma$ in (ii), yields a total of six such area-minimizing triangles, which are pedal relative to six pedal points, say, $P_1,....,P_6$. The main result of the paper is the fact that there exists a circle which contains all six points.

2014

Abstract: In this brief work, the authors confirm the existing formulae for the perimeter and area of a circle using a different approach.

The College Mathematics Journal, 2018

There are known constructions for some regular polygons, using only straightedge and compass, but not for all polygons-the Gauss-Wantzel Theorem states precisely which ones can be constructed. The constructions differ greatly from one polygon to the other. There are, however, general processes for determining the side of the n-gon (approximately, but sometimes with great precision), which we describe in this paper. We present a joint mathematical analysis of the so-called Bion and Tempier approximation methods, comparing the errors and trying to explain why these constructions would work at all.

Circling the Square with Straightedge and Compass in Euclidean Geometry, 2024

Abstract - There are three classical problems remaining from ancient Greek mathematics that are extremely influential in the development of Geometry. They are Trisecting An Angle, Squaring The Circle, and Doubling The Cube problems. The Squaring The Circle problem is solved accurately and is published in the International Journal Of Mathematics Trends And Technology (Volume 69, June 2023). Upstream from this method of exactly “Squaring The Circle”, one can conversely/inversely, deduce to get a new Mathematical challenge "CIRCLING THE SQUARE" with a straightedge & a compass in Euclidean Geometry. This study idea came from the exact solution “Squaring The Circle by Straightedge & compass in Euclidean Geometry”, published by IJMTT in June 2023 at https://ijmttjournal.org/Volume-69/Issue-6/IJMTTV69I6P506.pdf for this ancient Greek Geometry problem. In this research, the ANALYSIS method is adopted to prove the process of solving this new challenge problem, which has not existed in the Mathematics field till today. The process is an inverse/converse solution solving the ancient Greek Geometry challenge problem of “Squaring The Circle”, using a straightedge & a compass. I hereby commit that this is my own personal research project. Keywords - Circling the square, Circulating square, Circle mature of the square, Make square circled, Find circle area same as a square, Make a square rounded.

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...

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.