Paradigmatic View on the Concept of World Science Volume 1
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Polygon Review and Puzzlers in the Above, Those Are Names to the Polygons: Fill in the Blank Parts. Names: Number of Sides
Polygon review and puzzlers ÆReview to the classification of polygons: Is it a Polygon? Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Polygon Not a Polygon Not a Polygon (straight sides) (has a curve) (open, not closed) Regular polygons have equal length sides and equal interior angles. Polygons are named according to their number of sides. Name of Degree of Degree of triangle total angles regular angles Triangle 180 60 In the above, those are names to the polygons: Quadrilateral 360 90 fill in the blank parts. Pentagon Hexagon Heptagon 900 129 Names: number of sides: Octagon Nonagon hendecagon, 11 dodecagon, _____________ Decagon 1440 144 tetradecagon, 13 hexadecagon, 15 Do you see a pattern in the calculation of the heptadecagon, _____________ total degree of angles of the polygon? octadecagon, _____________ --- (n -2) x 180° enneadecagon, _____________ icosagon 20 pentadecagon, _____________ These summation of angles rules, also apply to the irregular polygons, try it out yourself !!! A point where two or more straight lines meet. Corner. Example: a corner of a polygon (2D) or of a polyhedron (3D) as shown. The plural of vertex is "vertices” Test them out yourself, by drawing diagonals on the polygons. Here are some fun polygon riddles; could you come up with the answer? Geometry polygon riddles I: My first is in shape and also in space; My second is in line and also in place; My third is in point and also in line; My fourth in operation but not in sign; My fifth is in angle but not in degree; My sixth is in glide but not symmetry; Geometry polygon riddles II: I am a polygon all my angles have the same measure all my five sides have the same measure, what general shape am I? Geometry polygon riddles III: I am a polygon. -
Formulas Involving Polygons - Lesson 7-3
you are here > Class Notes – Chapter 7 – Lesson 7-3 Formulas Involving Polygons - Lesson 7-3 Here’s today’s warmup…don’t forget to “phone home!” B Given: BD bisects ∠PBQ PD ⊥ PB QD ⊥ QB M Prove: BD is ⊥ bis. of PQ P Q D Statements Reasons Honors Geometry Notes Today, we started by learning how polygons are classified by their number of sides...you should already know a lot of these - just make sure to memorize the ones you don't know!! Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Undecagon 12 Dodecagon 13 Tridecagon 14 Tetradecagon 15 Pentadecagon 16 Hexadecagon 17 Heptadecagon 18 Octadecagon 19 Enneadecagon 20 Icosagon n n-gon Baroody Page 2 of 6 Honors Geometry Notes Next, let’s look at the diagonals of polygons with different numbers of sides. By drawing as many diagonals as we could from one diagonal, you should be able to see a pattern...we can make n-2 triangles in a n-sided polygon. Given this information and the fact that the sum of the interior angles of a polygon is 180°, we can come up with a theorem that helps us to figure out the sum of the measures of the interior angles of any n-sided polygon! Baroody Page 3 of 6 Honors Geometry Notes Next, let’s look at exterior angles in a polygon. First, consider the exterior angles of a pentagon as shown below: Note that the sum of the exterior angles is 360°. -
Volume 2 Shape and Space
Volume 2 Shape and Space Colin Foster Introduction Teachers are busy people, so I’ll be brief. Let me tell you what this book isn’t. • It isn’t a book you have to make time to read; it’s a book that will save you time. Take it into the classroom and use ideas from it straight away. Anything requiring preparation or equipment (e.g., photocopies, scissors, an overhead projector, etc.) begins with the word “NEED” in bold followed by the details. • It isn’t a scheme of work, and it isn’t even arranged by age or pupil “level”. Many of the ideas can be used equally well with pupils at different ages and stages. Instead the items are simply arranged by topic. (There is, however, an index at the back linking the “key objectives” from the Key Stage 3 Framework to the sections in these three volumes.) The three volumes cover Number and Algebra (1), Shape and Space (2) and Probability, Statistics, Numeracy and ICT (3). • It isn’t a book of exercises or worksheets. Although you’re welcome to photocopy anything you wish, photocopying is expensive and very little here needs to be photocopied for pupils. Most of the material is intended to be presented by the teacher orally or on the board. Answers and comments are given on the right side of most of the pages or sometimes on separate pages as explained. This is a book to make notes in. Cross out anything you don’t like or would never use. Add in your own ideas or references to other resources. -
Approximate Construction of a Regular Nonagon in Albrecht Dürer's ``Painter's Manual''
Approximate construction of a regular nonagon in Albrecht D¨urer’s Painter’s Manual: where had it come from? In Mathematical Cranks, in the chapter “Nonagons, regular”, Underwood Dudley gives an approximate straightedge and compass construction of a regular nonagon:1 Nonagons follow as corollaries from trisections, being easily made by trisecting 120◦ angles, and it would be an odd crank indeed who would pass up a famous and general problem for an obscure and particular one. Nevertheless, nonagoners exist, and Figure 1 is a nonagon construction that was made independently of any trisection. On the circle, mark off |AB| B F D A O E C Figure 1: The approximate nonagon construction in Mathematical Cranks. and |AC| equal to |OC| , and draw arcs from O to A with centers at B and C, both with radius |OA|. Trisect OA at D, draw EF perpendicular to OA, and you have the side of the nonagon inscribed in the circle with radius |OE|. That is, the angle OEF is supposed to be 40◦, but it falls short by quite a bit since it measures only 39.6◦. Writing α = OEF, we have 1 √ √ tan 1 α = 35 − 3 3 , 2 2 and 1 √ √ α = 2 arctan 35−3 3 =39.594◦ . 2 When we step around the circle with this approximate nonagon’s side, the gap that remains after the nine steps is 3.65◦, so this is a rather crude approximation. If this construction was ever advertised by some nonagoner as his (or, most improb- ably, hers) own, and as an exact construction at that, then there is a very real chance 1The quotation is not quite verbatim: the angle EOF in the original is written here as OEF, and the reference to Figure 31 in the book is replaced with the reference to its re-creation in this essay. -
Polygons Introduction
POLYGONS SIMPLE COMPLEX STAR POLYGONS TYPES ELEMENTS NUMBER OF SIDES CONCAVE CONVEX NAMES 3- TRIANGLES REGULAR IRREGULAR 4-QUADRILATERALS 5-PENTAGON SIDES 6-HEXAGON 7-HEPTAGON CONSTRUCTION DIAGONAL 8-OCTAGON 9-ENNEAGON VERTICES 10-DECAGON .... INSCRIBED GIVEN OTHER ALTITUDE THE SIDE ITEMS Star polygons: It is a particular polygon case with a star shape, created out of linking together non consecutive vertices of a regular polygon. INSCRIBED VERSUS CIRCUMSCRIBED: Inscribed Polygons: They are polygons placed inside circles so all the vertices of the polygon are placed on the circle. Circumscribed polygons: They are polygons surrounding a circle, being its sides tangent to the circle inside. A- The square is inscribed in the circle while the circle is circunscribed to the square. A B B- The circle is inscribed in the triangle while the triangle is circumscribed to the triangle. Watch this short Youtube video that introduces you the polygons definition FLATLAND - Edwin A. Abbott 1838-1926 and some names http://youtu.be/LfPDFGvGbqk 3. CONCERNING THE INHABITANTS OF FLATLAND The greatest length or breadth of a full grown inhabitant of Flatland may be estimated at about eleven of your inches. Twelve inches may be regarded as a maximum. Our Women are Straight Lines. Our Soldiers and Lowest Class of Workmen are Triangles with two equal sides, each about eleven inches long, and a base or third side so short (often not exceeding half an inch) that they form at their vertices a very sharp and formidable angle. Indeed when their bases are of the most degraded type (not more than the eighth part of an inch in size), they can hardly be distinguished from Straight lines or Women; so extremely pointed are their vertices. -
Geometry in Design Geometrical Construction in 3D Forms by Prof
D’source 1 Digital Learning Environment for Design - www.dsource.in Design Course Geometry in Design Geometrical Construction in 3D Forms by Prof. Ravi Mokashi Punekar and Prof. Avinash Shide DoD, IIT Guwahati Source: http://www.dsource.in/course/geometry-design 1. Introduction 2. Golden Ratio 3. Polygon - Classification - 2D 4. Concepts - 3 Dimensional 5. Family of 3 Dimensional 6. References 7. Contact Details D’source 2 Digital Learning Environment for Design - www.dsource.in Design Course Introduction Geometry in Design Geometrical Construction in 3D Forms Geometry is a science that deals with the study of inherent properties of form and space through examining and by understanding relationships of lines, surfaces and solids. These relationships are of several kinds and are seen in Prof. Ravi Mokashi Punekar and forms both natural and man-made. The relationships amongst pure geometric forms possess special properties Prof. Avinash Shide or a certain geometric order by virtue of the inherent configuration of elements that results in various forms DoD, IIT Guwahati of symmetry, proportional systems etc. These configurations have properties that hold irrespective of scale or medium used to express them and can also be arranged in a hierarchy from the totally regular to the amorphous where formal characteristics are lost. The objectives of this course are to study these inherent properties of form and space through understanding relationships of lines, surfaces and solids. This course will enable understanding basic geometric relationships, Source: both 2D and 3D, through a process of exploration and analysis. Concepts are supported with 3Dim visualization http://www.dsource.in/course/geometry-design/in- of models to understand the construction of the family of geometric forms and space interrelationships. -
TOPIC 11 Triangles and Polygons
TOPIC 11 Triangles and polygons We can proceed directly from the results we learned about parallel lines and their associated angles to the ideas concerning triangles. An essential question is a question whose answer is not obvious, but it is a vitally important question that we continue to ponder. Essential question: Why are triangles so important in the study of geometry? We will spend more time studying triangles than any other single topic. Why? Let’s review the definitions: 1. What is a triangle? 2. What is an isosceles triangle? 3. What is an equilateral triangle? 4. What is a scalene triangle? 5. What is a scalene triangle? 6. What is an acute triangle? 7. What is an obtuse triangle? 8. What is a right triangle? Topic 11 (Triangles and Polygons) page 2 What can we discover that is true about a triangle? One well-known result concerns the sum of the angles of a triangle. We will do this with a paper triangle in class … a sort of kinesthetic proof. In a more formal way, we will work through a written proof. Given: Δ ABC Prove: m< 1 + m< 2 + m< 3 = 180 Proof: 1. Draw ECD || AB 1. 2. < 2 ______ 2. 3. < 3 ______ 3. 4. < ____ is supp to < ACD 4. 5. m < ____ + m < ACD = 180 5. Def supp 6. m< 1 +m < ____ = m < ACD 6. angle addition postulate 7. m < 1 + m < ____ + m < ____= 180 7 . 8. m< 1 + m < 2 + m < 3 = 180 8. QED Is the parallel postulate necessary in this proof? (We use this as the parallel postulate: If corresponding angles are congruent, then lines are parallel.) As an implication of this fact, we accept as true the fact that there is only one line through a point which is parallel to a given line. -
REALLY BIG NUMBERS Written and Illustrated by Richard Evan Schwartz
MAKING MATH COUNT: Exploring Math through Stories Great stories are a wonderful way to get young people of all ages excited and interested in mathematics. Now, there’s a new annual book prize, Mathical: Books for Kids from Tots to Teens, to recognize the most inspiring math-related fiction and nonfiction books that bring to life the wonder of math in our lives. This guide will help you use this 2015 Mathical award-winning title to inspire curiosity and explore math in daily life with the youth you serve. For more great books and resources, including STEM books and hands-on materials, visit the First Book Marketplace at www.fbmarketplace.org. REALLY BIG NUMBERS written and illustrated by Richard Evan Schwartz Did you know you can cram 20 billion grains of sand into a basketball? Mathematician Richard Evan Schwartz leads readers through the GRADES number system by creating these sorts of visual demonstrations 6-8 and practical comparisons that help us understand how big REALLY WINNER big numbers are. The book begins with small, easily observable numbers before building up to truly gigantic ones. KEY MATH CONCEPTS The Mathical: Books for Kids from Tots to Teens book prize, presented by Really Big Numbers focuses on: the Mathematical Sciences Research • Connecting enormous numbers to daily things familiar to Institute (MSRI) and the Children’s students Book Council (CBC) recognizes the • Estimating and comparing most inspiring math-related fiction and • Having fun playing with numbers and puzzles nonfiction books for young people of all ages. The award winners were selected Making comparisons and breaking large quantities into smaller, by a diverse panel of mathematicians, easily understood components can help students learn in ways they teachers, librarians, early childhood never thought possible. -
CONVERGENCE of the RATIO of PERIMETER of a REGULAR POLYGON to the LENGTH of ITS LONGEST DIAGONAL AS the NUMBER of SIDES of POLYGON APPROACHES to ∞ Pawan Kumar B.K
CONVERGENCE OF THE RATIO OF PERIMETER OF A REGULAR POLYGON TO THE LENGTH OF ITS LONGEST DIAGONAL AS THE NUMBER OF SIDES OF POLYGON APPROACHES TO ∞ Pawan Kumar B.K. Kathmandu, Nepal Corresponding to: Pawan Kumar B.K., email: [email protected] ABSTRACT Regular polygons are planar geometric structures that are used to a great extent in mathematics, engineering and physics. For all size of a regular polygon, the ratio of perimeter to the longest diagonal length is always constant and converges to the value of 휋 as the number of sides of the polygon approaches to ∞. The purpose of this paper is to introduce Bishwakarma Ratio Formulas through mathematical explanations. The Bishwakarma Ratio Formulae calculate the ratio of perimeter of regular polygon to the longest diagonal length for all possible regular polygons. These ratios are called Bishwakarma Ratios- often denoted by short term BK ratios- as they have been obtained via Bishwakarma Ratio Formulae. The result has been shown to be valid by actually calculating the ratio for each polygon by using corresponding formula and geometrical reasoning. Computational calculations of the ratios have also been presented upto 30 and 50 significant figures to validate the convergence. Keywords: Regular Polygon, limit, 휋, convergence INTRODUCTION A regular polygon is a planar geometrical structure with equal sides and equal angles. For a 푛(푛−3) regular polygon of 푛 sides there are diagonals. Each angle of a regular polygon of n sides 2 푛−2 is given by ( ) 180° while the sum of interior angles is (푛 − 2)180°. The angle made at the 푛 center of any polygon by lines from any two consecutive vertices (center angle) of a polygon of 360° sides 푛 is given by . -
2020 James S. Rickards Fall Invitational Geometry Individual for All Questions, Answer Choice (E) NOTA Means That None of the Given Answers Is Correct
2020 James S. Rickards Fall Invitational Geometry Individual For all questions, answer choice (E) NOTA means that none of the given answers is correct. Good Luck! 1. Given a certain statement is true, which of the following must be true? (A) Its converse (B) Its inverse (C) Its contrapositive (D) Depends on the statement (E) NOTA 2. The vertices of a 2020-gon are labeled in order as A1, A2,..., A2020. The diagonals of that polygon are drawn. Justin erases all of the diagonals that contain A19 or A2019. How many diagonals are left? (A) 2035153 (B) 2035155 (C) 2033134 (D) 2033136 (E) NOTA 3. What is the area of the triangle bounded by the lines: y = x − 4, y = 2x − 1, y = 3x − 2? (A) 3 (B) 4 (C) 6 (D) 8 (E) NOTA 4. Akhil's favorite formula is the Brahmagupta formula. What is the area of a cyclic quadrilateral whose side lengths are 4,5,7, and 10? (A) 72 (B) 36 (C) 78 (D) 1296 (E) NOTA 5. Triangle ABC is an isosceles triangle. If the length of AB is 20 units and angle C is equal to 120 degrees. What is the sum of the numerical valuesp of the area and perimeterp of the triangle? 40 3 140 3 (A) 180 (B) + 20 (C) + 20 (D) Not possible (E) NOTA 3 3 6. Sruthi decidesp to inscribe a circle into the triangle from the previous question. What is the areap of the inscribed circle? 10 − 5 3 p 100π (175 − 100 3)π (A) (B) (700 − 400 3)π (C) (D) (E) NOTA 4 3 16 7. -
On the Projection of the Regular Polytope { 5, 3, 3} Into a Regular Triacontagon
ON THE PROJECTION OF THE REGULAR POLYTOPE { 5, 3, 3} INTO A REGULAR TRIACONTAGON B. L. Chilton (received November 29, 1963) 1. Introduction. The polytope {5,3,3} is one of the six convex regular four-dimensional polytope s, and in some ways is the most complicated of the six, being bounded by 120 dodecahedra. The symbol { p, q, r} to denote a regular figure in four dimen sions originated with L. Schlafli: p is the number of edges bounding each face, q is the number of edges of a cell which meet at a vertex of this ceil, and r is the number of ceils that share each edge. The Schlâfli symbol can be generalized to denote regular figures in any number of dimensions; in parti cular, a regular p-gon is symbolized by { p} , and a dodecahedron by {5,3}. It is possible to project the 120-cell (as the polytope { 5, 3,3} is "popularly" called) onto a plane so that the projection thus formed is bounded by a regular 30-gon. This polygon arises as the plane projection of a Petrie polygon of the 120-cell. The Petrie polygon of an n-dimensional regular polytope (Coxeter [2], p. 223) is defined inductively as a skew polygon such that any n - 1 consecutive sides, but no n, belong to a Petrie polygon of a cell. The induction begins with the case n = 3, when Ha Petrie polygon of a cell" has to be replaced by na face". Because of the way in which the projection arises, its symmetry group will be that of the regular triacontagon, namely, B^n* the dihedral group of order 60. -
Geometry and Mensuration IV.7 11 Geometry and Mensuration
Geometry and Mensuration IV.7 11 Geometry and Mensuration The chapters on geometry and mensuration have their own The following is a comprehensive collection of formulae share of questions in the CAT and other MBA entrance ex- based on two-dimensional figures. The student is advised aminations. For doing well in questions based on this chapter, to remember the formulae in this chapter so that he is able the student should familiarise himself/herself with the to solve all the questions based on this chapter. basic formulae and visualisations of the various shapes of solids and two-dimensional figures based on this chapter. THEORY The following is a comprehensive collection of for- mulae based on two-dimensional and three-dimensional Basic Conversions figures: For the purpose of this chapter we have divided the A. 1 m = 100 cm = 1000 mm B. 1 m = 39.37 inches theory in two parts: 1 km = 1000 m 1 mile = 1760 yd ∑ Part I consists of geometry and mensuration of two- = 5/8 miles = 5280 ft dimensional figures 1 inch = 2.54 cm 1 nautical mile (knot) ∑ Part II consists of mensuration of three-dimensional = 6080 ft figures. C. 100 kg = 1 quintal D. 1 litre = 1000 cc 10 quintal = 1 tonne 1 acre = 100 sq m = 1000 kg PART I: GEOMETRY 1 kg = 2.2 pounds 1 hectare = 10000 sq m (approx.) INTRODUCTION TYPES OF ANGLES Geometry and Mensuration are important areas in the CAT examination. In the Online CAT, the Quantitative Aptitude Basic Definitions section has consisted of an average of 15–20% questions from these chapters.