Mathematics
Polygons
Polygons are closed geometric shapes with straight sides. They are defined by the number of sides they have, with a minimum of three sides. Common polygons include triangles, quadrilaterals, pentagons, and hexagons. The interior angles of a polygon can be calculated using the formula (n-2) * 180°, where n is the number of sides.
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No longer available |Learn moreCollege Geometry
A Unified Approach
- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 5 Polygon and Circle Polygon An assortment of Polygons In geometry a polygon is traditionally a plane figure that is bounded by a closed path or circuit , composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides , and the points where two edges meet are the polygon's vertices or corners . An n -gon is a polygon with n sides. The interior of the polygon is sometimes called its body . A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The word polygon derives from the Greek πολύς (many) and γωνία (gōnia), meaning knee or angle. Today a polygon is more usually understood in terms of sides. Usually two edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the line segments will be considered parts of a single edge. The basic geometrical notion has been adapted in various ways to suit particular purposes. For example in the computer graphics (image generation) field, the term polygon has taken on a slightly altered meaning, more related to the way the shape is stored and manipulated within the computer. ________________________ WORLD TECHNOLOGIES ________________________ Classification Number of sides Convexity Polygons may be characterised by their degree of convexity: • Convex : any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. In other words, all its interior angles are less than 180°. • Non-convex : a line may be found which meets its boundary more than twice. In other words, it contains at least one interior angle with a measure larger than 180°. • Simple : the boundary of the polygon does not cross itself. All convex Polygons are simple. • Concave : Non-convex and simple. • Star-shaped : the whole interior is visible from a single point, without crossing any edge.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 5 Polygon and Circle Polygon An assortment of Polygons In geometry a polygon is traditionally a plane figure that is bounded by a closed path or circuit , composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides , and the points where two edges meet are the polygon's vertices or corners . An n -gon is a polygon with n sides. The interior of the polygon is sometimes called its body . A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The word polygon derives from the Greek πολύς (many) and γωνία (gōnia), meaning knee or angle. Today a polygon is more usually understood in terms of sides. Usually two edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the line segments will be considered parts of a single edge. The basic geometrical notion has been adapted in various ways to suit particular purposes. For example in the computer graphics (image generation) field, the term polygon has taken on a slightly altered meaning, more related to the way the shape is stored and manipulated within the computer. ________________________ WORLD TECHNOLOGIES ________________________ Classification Number of sides Convexity Polygons may be characterised by their degree of convexity: • Convex : any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. In other words, all its interior angles are less than 180°. • Non-convex : a line may be found which meets its boundary more than twice. In other words, it contains at least one interior angle with a measure larger than 180°. • Simple : the boundary of the polygon does not cross itself. All convex Polygons are simple. • Concave : Non-convex and simple. • Star-shaped : the whole interior is visible from a single point, without crossing any edge.
eBook - PDFCollege Geometry
A Unified Development
- David C. Kay(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
161 4 Quadrilaterals,. Polygons,.and.Circles The fundamental properties of Polygons and circles commonly studied in elementary geometry may be derived axiomatically with relative ease. The purpose of the present chapter is to explore that derivation briefly in the context of unified geometry. The polygon has an important spe-cial case in the quadrilateral—the object of discussion for the opening sections. 4 .1. Quadrilaterals Of great importance in geometry, topology, and analysis is the concept of a polygonal path —a set consisting of those segments ( sides ) joining a sequence of distinct points ( vertices ). Our focus at first is on closed polyg-onal paths of order four (four-sided polygonal paths where the first and last vertices coincide). We might note that, if we are given four points, there are precisely three distinct closed polygonal paths of order four hav-ing those points as vertices, and the location of those points relative to each other falls into two classes: (1) when the paths determined by the points are not self-intersecting (called simple ), as shown in Figure 4.1, and (2) when some paths self-intersect, as shown in Figure 4.2. Since a closed polygonal path can enclose a well-defined interior only if it is not self-intersecting, we are only interested in paths that do not self-intersect. In the two figures there are only four of these, which are called quadrilat-erals : all three in Figure 4.1 and one in Figure 4.2. Figure 4 .1 Simple closed paths of order four. 162 College Geometry: A Unified Development Figure 4 .2 Closed paths of order four. Definition 1 Let P 0 , P 1 , P 2 , P 3 , P 4 be 5 distinct points in the plane such that for each integer i and j in the range [0, 4], P i P j < α . A polygo-nal path of order four joining P 0 and P 4 is the set [ ] P PP P P P P PP P P P P 0 1 2 3 4 0 1 1 2 2 3 3 4 = ∪ ∪ ∪ with the given points called its vertices , and the segments joining them, its sides (Figure 4.3).
eBook - ePub- Steven J. Janke(Author)
- 2014(Publication Date)
- Wiley(Publisher)
Chapter 6 Polygons and PolyhedraSince lines and planes are fundamental to geometry, shapes bounded by lines and planes at least have access to the center stage in computer graphics. Most modeling efforts, no matter how they begin, usually end up with a vast assortment of triangles because this shape is guaranteed to be planar even when the vertices are points in space. There are online repositories of models composed entirely of very large sets (thousands) of triangles. Understanding the geometry of triangles and how to efficiently use them in computation is particularly important in graphics. More general Polygons arise when constructing complex objects (polyhedra) and when projecting those objects to find shadows. The geometry of both Polygons and polyhedra is the key to much of graphics and gives rise to a wide range of mathematical tools.6.1 Triangles
Triangles are well studied in elementary geometry, and with vector geometry we have the tools to calculate most of what we need in graphics: side lengths, angles, and areas. One key problem is to determine whether a point of intersection is inside a triangle in space. Although elementary tools can suffice, there is always a quest to find better ways to express the problem in the hope of improving computational efficiency. Barycentric coordinates offer a different view of triangle geometry and thereby lead to some nice algorithms.6.1.1 Barycentric Coordinates
Recall from Chapter 2 (Section 2.1.1) that addition of points cannot be defined uniquely. However, if we take an affine combination , where , then we do get a well-defined point. This is why we can represent points on a line as . The numbers and are called barycentric coordinates and determine the location of a point relative to the reference points and
No longer available |Learn more- Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
CHAPTER OUTLINE 8.1 Area and Initial Postulates 8.2 Perimeter and Area of Polygons 8.3 Regular Polygons and Area 8.4 Circumference and Area of a Circle 8.5 More Area Relationships in the Circle ■ PERSPECTIVE ON HISTORY: Sketch of Pythagoras ■ PERSPECTIVE ON APPLICATIONS: Another Look at the Pythagorean Theorem ■ SUMMARY Chapter 8 © Glowimages/Getty Images 341 Powerful! The unique shape and the massive size of the Pentagon in Washington, D.C., manifest the notion of strength. In this chapter, we introduce the concept of area. The area of an enclosed plane region is a measure of size that has applications in construction, farming, real estate, and more. Some of the units that are used to measure area include the square inch and the square centimeter. While the areas of square and rectangular regions are generally easily calculated, we will also develop formulas for the areas of less common polygonal regions. In particular, Section 8.3 is devoted to calculating the areas of regular Polygons, such as the Pentagon shown in the photograph. Many real-world applications of the area concept are found in the examples and exercise sets of this chapter. Areas of Polygons and Circles Additional video explanations of concepts, sample problems, and applications are available on DVD. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 342 CHAPTER 8 ■ AREAS OF Polygons AND CIRCLES Unless otherwise noted, all content on this page is © Cengage Learning. Lines are said to be one-dimensional because we can measure only the length of a line segment.
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
4 Geometry Focal Points- Undefined Terms
- Angles
- Simple Closed Curves, Regions, and Polygons
- Circles
- Constructions
- Third Dimension
- Coordinate Geometry
- Transformations and Symmetry
You might be surprised about how many real-life concepts are included in the study of geometry. Young children experiment with ideas such as over versus under, first versus last, right versus left, and between, without realizing that they are studying important mathematics concepts. Additionally, early attempts at logic, even the common everyone else gets to do it arguments that are so popular with children, are geometry topics. A cube is a three-dimensional object with six congruent faces and eight vertices—often called a block. You may also use the term block to identify a prism that is not a cube, but, although you may not think of it very often, you can probably identify the differences between a cube and a prism that is not a cube, as shown in Fig. 4.1 .Fig. 4.1.In this chapter, you will review, refine, and perhaps, extend your understanding of geometry. When Euclid completed a series of 13 books called the Elements in 300 BC , he provided a logical development of geometry that is unequaled in our history and is the foundation of our modern geometry study. Geometry is a dynamic, growing, and changing body of intuitive knowledge. We will let you explore conjectures and provide opportunities for you to create informal definitions. There will be some reliance on terms and previous knowledge, especially when we get to standard formulas.Undefined TermsSome fundamental concepts in geometry defy definition. If we try to define point, space, line, and plane, then we find ourselves engaging in circular (flawed) logic. The best we can do is accept these fundamental concepts as building blocks and try to explain them.A fixed location is called a point, which is a geometric abstraction that has no dimension, only position. We often use a tiny round dot as a representation of a point. As the series of dots in Fig. 4.2 get smaller and smaller, we observe that the dimensions are diminishing—but any dot that we can see has some dimension, even the period at the end of this sentence. The fact that a point has no physical existence does not limit its usefulness, either in geometry or everyday activities. Although a dot covers an infinite number of points, it represents the approximate location of a distinct point so well that we forget the difference and freely identify the dot as a pinpointed location. In mathematics, we label the point represented by a dot with a printed capital letter, for example, the points in Fig. 4.3 . The set of all possible fixed locations is called space
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
351 ©John Peter Photography / Alamy Stock Photo. 8 Areas of Polygons and Circles Elegant! The Pittman Dowell Residence, which lies 15 miles north of Los Angeles, has the rather improbable shape of a heptagon. In this chapter, we introduce the concept of area of a region; for instance, the dwelling shown above has an interior that measures 3200 square feet. The area of an enclosed plane region is a measure of size that has applications in construction, farming, real estate, and more. Some of the units that are used to measure area include the square inch and the square centimeter. While the areas of square and rectangular regions are generally easily calculated, we will also develop formulas for the areas of less common polygonal regions. In particular, Section 8.3 is devoted to calculating the areas of regular Polygons, such as the one shown in the photograph. We also develop the formula for the area of a circle (circular region) in Section 8.4. Many real-world applications of the area concept are found in the examples and exercise sets of this chapter. CHAPTER OUTLINE 8.1 Area and Initial Postulates 8.2 Perimeter and Area of Polygons 8.3 Regular Polygons and Area 8.4 Circumference and Area of a Circle 8.5 More Area Relation- ships in the Circle ■ PERSPECTIVE ON HISTORY: Sketch of Pythagoras ■ PERSPECTIVE ON APPLICATIONS: Another Look at the Pythagorean Theorem ■ SUMMARY Photo : Iwan Baan 352 CHAPTER 8 ■ AREAS OF Polygons AND CIRCLES KEY CONCEPTS Plane Region Square Unit Area Postulates Area of a Rectangle, a Parallelogram, and a Triangle Altitude and Base of a Parallelogram and a Triangle 8.1 Area and Initial Postulates A line is said to be one-dimensional because the only measure related to a line is the length of a line segment. Of course, such measures are only good approximations. A line segment is measured in linear units such as inches, centimeters, or yards.
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Then, through talking about their models, the students are able to learn and use new vocabulary in a meaningful way (Koester, 2003). Common Core – Grade 6 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface areas of these figures. Apply these techniques in the context of solv- ing real-world and mathematical problems. 624 Chapter 12 Geometric Shapes Since there are infinitely many types of Polygons to use as the bases, there are infinitely many types of prisms. Pyramids are polyhedra formed by using a polygon for the base and a point not in the plane of the base, called the apex, that is connected with line segments to each vertex of the base. Figure 12.94 shows several pyramids, named according to the type of polygon forming the base. Pyramids whose bases are regular Polygons fall into two categories. Those whose lateral faces are isosceles triangles are called right regular pyramids. Otherwise, they are oblique regular pyramids. Right triangular pyramid Right square pyramid Oblique square pyramid Right pentagonal pyramid Figure 12.94 Polyhedra with regular Polygons for faces have been studied since the time of the ancient Greeks. A regular polyhedron is one in which all faces are identical regular polygonal regions and all dihedral angles have the same measure. The ancient Greeks were able to show that there are exactly five regular convex polyhedra, called the Platonic solids. They are analyzed in Table 12.10, according to number of faces, ver- tices, and edges, and shown in Figure 12.95. An interesting pattern in Table 12.10 is that F V E + = + 2 for all five regular polyhedra. That is, the number of faces plus ver- tices equals the number of edges plus 2. This result, known as Euler’s formula, holds for all convex polyhedra, not just regular polyhedra. For example, verify Euler’s formula for each of the polyhedra in Figures 12.92, 12.93, and 12.94.
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Stop for a moment and think of examples, both natural and human-made objects. Then read on. . . . All of the figures in Figure 8.90 are Polygons. • The stop sign is an octagon—an eight-sided polygon. • The common nut has a hexagonal shape—a six-sided polygon. • The Pentagon in Washington has five sides. Let us examine a few important aspects of Polygons with more than four sides. First, we distinguish between regular and nonregular Polygons. What do you think a regular pentagon or a regular hexagon is? How might we define it? Think about this and write down your thoughts before reading on. . . . A regular polygon is one in which all sides have the same length and all interior angles have the same measure. What do we call a regular quadrilateral? What about a regular triangle? Is it possible for a regular polygon to be concave? Think about these questions before reading on. . . . Rhombus Square Rectangle ©Frontpage/Shutterstock.com ©Szasz-Fabian Jozsef/Shutterstock.com ©Frontpage/Shutterstock.com Figure 8.90 Figure 8.89 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 468 CHAPTER 8 Geometry as Shape Unless otherwise noted, all content on this page is © Cengage Learning CLASSROOM CONNECTION Grade 4 What are your answers for Question 3? From Everyday Mathematics, Grade 4: The University of Chicago School Mathematics Project: Student Math Journal, Volume 1, by Max Bell et al., Lesson 3-1, p. 54. Reprinted by permission of The McGraw-Hill Companies, Inc. Copyright 2016 Cengage Learning. All Rights Reserved.
eBook - PDF- Paul Heckbert(Author)
- 1994(Publication Date)
- Morgan Kaufmann(Publisher)
♦ IO Polygons and Polyhedra This part of the book contains five Gems on Polygons and three on polyhedra. Polygons and polyhedra are the most basic and popular geometric building blocks in computer graphics. 1.1. Centroid of a Polygon, by Gerard Bashein and Paul R. Detmer. Gives formulas and code to find the centroid (center of mass) of a polygon. This is useful when simulating Newtonian dynamics. Page 3. 1.2. Testing the Convexity of a Polygon, by Peter Schorn and Frederick Fisher. Gives an algorithm and code to determine if a polygon is convex, non-convex (concave but not convex), or non-simple (self-intersecting). For many polygon operations, faster algorithms can be used if the polygon is known to be convex. This is true when scan converting a polygon and when determining if a point is inside a polygon, for instance. Page 7. 1.3. An Incremental Angle Point in Polygon Test, by Kevin Weiler. 1.4. Point in Polygon Strategies, by Eric Haines. Provide algorithms for testing if a point is inside a polygon, a task known as point inclusion testing in computational geometry. Point-in-polygon testing is a basic task when ray tracing polygonal models, so these methods are useful for 3D as well as 2D graphics. Weiler presents a single algorithm for testing if a point lies in a concave polygon, while Haines surveys a number of algorithms for point inclusion testing in both convex and concave Polygons, with empirical speed tests and practical optimizations. Pages 16 and 24. 1 2 0 Polygons and Polyhedra 1.5. Incremental Delaunay Triangulation, by Dani Lischinski. Gives some code to solve a very important problem: finding Delaunay triangulations and Voronoi diagrams in 2D. These two geometric constructions are useful for trian-gular mesh generation and for nearest neighbor finding, respectively. Triangular mesh generation comes up when doing interpolation of surfaces from scattered data points, and in finite element simulations of all kinds, such as radiosity.
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