Regular Polygon

12 Key excerpts on "Regular Polygon"

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  eBook - ePub

  Tessellations

  Mathematics, Art, and Recreation

  • Robert Fathauer(Author)
  • 2020(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  Figure 2.1. Three examples of things that are not tiles. (a) A collection of unconnected shapes; (b) two shapes touching at a point; (c) a shape with a hole in it.

  A polygon is a closed plane figure made up of straight-line segments . For a polygonal tile, the individual segments are referred to as edges , and the points where two edges meet are referred to as corners (Figure 2.1 ). Examples of tiles that are not polygonal are shown in Figure 2.3 .

  Figure 2.2. A polygonal tile has a boundary consisting of edges and corners.

  Figure 2.3. Besides being polygons, tiles can have no straight lines or a mixture of straight and curved lines.

  Angles

  For a polygonal tile, the angle between two adjacent edges inside the tile is called the interior angle . The angle between the same two edges outside the tile is called the exterior angle . A full revolution measures 360° (Figure 2.4 ). As a result, for any two adjacent edges of a tile, the interior and exterior angles always sum to 360°. Some common angles encountered in tessellations, particularly those of Regular Polygons, are shown in Figure 2.5 . Each angle is specified in three different ways: in degrees, as a fraction of π, and as a fraction of a full revolution. A full revolution equals 360° or 2π. These comments about angles pertain to the Euclidean plane , the two-dimensional geometric space students learn about in K–12 education. Unless otherwise stated, tessellations in this book should be understood to live in this space.

  Figure 2.4. The interior and exterior angles between two adjacent edges sum to 360°.

  Figure 2.5. Common angles in tessellations.

  Vertices and edge-to-edge tessellations

  A point at which three or more tiles meet is called a vertex (Figure 2.6 ). The sums of the interior angles of the tiles meeting at a vertex must be 360°, which is a full revolution. The number of tiles meeting at a vertex is the

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  Technical Mathematics with Calculus

  • Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  6-19. The points where the sides meet are called vertices. We say that the sides of a polygon are equal if their measures (lengths) are equal. If all of the sides and angles of a polygon are equal, it is called a Regular Polygon, as in Fig. 6-20. The perimeter of a polygon is simply the sum of its sides. Modern definitions of plane figures don’t include the interior as part of the figure. Thus in Fig. 6-20, point A is not on the square, while point B is on the square. The interior is referred to as a region. Regular Heptagon Regular Nonagon Regular Decagon Regular Octagon Interior angle Vertex Side FIGURE 6-19 A polygon Equilateral triangle Square Regular Hexagon Regular Pentagon B A FIGURE 6-20 Some Regular Polygons Sum of Interior Angles A polygon of n sides has n interior angles, such as those shown in Fig. 6-21. Their sum is equal to the following: Interior Angles of a Polygon Sum of angles = (n - 2)180° 112 99° 123° 62° 278° 226° 43° θ FIGURE 6-21 ◆◆◆ Example 4: Find angle θ in Fig. 6-21. Solution: The polygon shown (a heptagon) in Fig. 6-21 has seven sides, so n = 7. Using our equation for sum of interior angles (Eq. 112), let’s take a step approach. TABLE 6-2 Calculating the Interior Angle of a Polygon Step # Instruction Solution Step 1 Calculate what the sum of the interior angles should be. Sum of angles = (7 - 2) 180° = 900° Step 2 Add the six interior angles that we are given. 278° + 62° + 123° + 99° + 226° + 43° = 831° Step 3 Subtract the sum of the six interior angles we are given from what the sum of the interior angles should be. θ = 900° - 831° = 69° ◆◆◆ 131 Section 6–3 ◆ Triangles Exercise 2 ◆ Polygons 1. Calculate the sum of interior angles for the following Regular Polygons: (a) Triangle (b) Quadrilateral (c) Pentagon (d) Hexagon (e) Heptagon (f) Octagon (g) Nonagon (h) Decagon 2. Find the missing interior angle for the following irRegular Polygons: 6–3 Triangles Triangles are polygons with three sides.

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  Number, Shape, & Symmetry

  An Introduction to Number Theory, Geometry, and Group Theory

  • Diane L. Herrmann, Paul J. Sally Jr.(Authors)
  • 2012(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  Definition. A regular polyhedron is a convex polyhedron whose faces are congruent Regular Polygons with the same number of faces meeting at each vertex. As with polygons, first we will classify the regular polyhedra and then we will study their symmetries. The most familiar regular polyhedron is the cube (Figure 13.1). A cube has faces that are squares, with three meeting at each vertex. There are six faces, twelve edges, and eight vertices. There are also regular polyhedra whose faces are equilateral triangles. The simplest is one made of just four equilateral triangles as faces, called a regular tetrahedron (Figure 13.2). Three of the triangles meet at each vertex. The regular tetrahedron has four faces, four vertices, and six edges. How do we create a model of the tetrahedron? One way to begin is to surround a vertex with three equilateral triangles. We start by drawing a diagram with three equilateral triangles sharing a vertex, as shown in Figure 13.3. If we cut out this figure and fold it along the dotted lines, we get one corner of a regular tetrahedron. To finish the model, we need only to add the last face, as shown in Figure 13.4. Figure 13.1. Cube. Figure 13.2. Regular tetrahedron. 13.1 . Regular Polyhedra 253 Figure 13.3. Building a tetrahedron model. Figure 13.4. A flat tetrahedron. For a different regular polyhedron with four equilateral triangles at each vertex, we draw the diagram in Figure 13.5, with four equilateral triangles sharing a vertex. Figure 13.5. An-other flat pyramid. If we cut out this figure and fold it along the dotted lines, we get a square pyramid (Figure 13.6), which is one corner of a regular octahedron . Figure 13.6. Square pyramid. To complete the regular octahedron (Figure 13.7), we make another square pyramid and join the two along their square bottoms. The regular octahedron has eight faces, six vertices, and twelve edges. Note that in the regular octahedron, four triangles meet at each vertex.

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  Mathematics for Elementary School Teachers

  • Tom Bassarear, Meg Moss(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  54. Reprinted by permission of The McGraw-Hill Companies, Inc. 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. Unless otherwise noted, all content on this page is © Cengage Learning Section 8.2 Two-Dimensional Figures 469 A regular quadrilateral is called a square. A regular triangle is called an equilateral triangle. A Regular Polygon cannot be concave. A critical reader might be wondering whether you can have a polygon—let’s say a pentagon—where all the sides have the same length but not all the angles have the same measure. And what about the converse: Can you have a pentagon where all the angles have the same measure but not all the sides have the same length? What do you think? This will be left as an exercise. INVESTIGATION 8.2k Summing Triangle Angles and Those of Other Polygons Let’s explore an interesting concept with polygons. Just by knowing the number of angles that a polygon has, we can know the answer to finding the sum when we add all of the measures of the angles of that polygon. Let’s see how. We will get started together, and then we will invite you to take some time to see the pattern and develop the formula on your own. You can choose whether to follow using paper and pencil method below, or the Geogebra (free software) method. Initially, use the one that is most aligned with your learning style. We also encourage you to work through both tools, as understanding both will support you in understanding more learning styles of your future students.

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  College 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.

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  College Mathematics

  • (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.

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  8.3 ■ Regular Polygons and Area 377 9. In a particular type of Regular Polygon, the length of the radi- us is exactly the same as the length of a side of the polygon. What type of Regular Polygon is it? 10. In a particular type of Regular Polygon, the length of the apo- them is exactly one-half the length of a side. What type of Regular Polygon is it? 11. In one type of Regular Polygon, the measure of each interior angle 1 I 5 (n 2 2)180uni00B0 n 2 is equal to the measure of each central angle. What type of Regular Polygon is it? 12. If the area (A 5 1 2 aP) and the perimeter of a Regular Polygon are numerically equal, find the length of the apothem of the Regular Polygon. 13. Find the area of a square with apothem a 5 3.2 cm and perimeter P 5 25.6 cm. 14. Find the area of an equilateral triangle with apothem a 5 3.2 cm and perimeter P 5 19.2 !3 cm. 15. Find the area of an equiangular triangle with apothem a 5 4.6 in. and perimeter P 5 27.6 !3 in. 16. Find the area of a square with apothem a 5 8.2 ft and perimeter P 5 65.6 ft. In Exercises 17 to 30, use the formula A 5 1 2 aP to find the area of the Regular Polygon described. 17. Find the area of a regular pentagon with an apothem of length a 5 5.2 cm and each side of length s 5 7.5 cm. 18. Find the area of a regular pentagon with an apothem of length a 5 6.5 in. and each side of length s 5 9.4 in. 19. Find the area of a regular octagon with an apothem of length a 5 9.8 in. and each side of length s 5 8.1 in. 20. Find the area of a regular octagon with an apothem of length a 5 7.9 ft and each side of length s 5 6.5 ft. 21. Find the area of a regular hexagon whose sides have length 6 cm. 22. Find the area of a square whose apothem measures 5 cm. 23. Find the area of an equilateral triangle whose radius measures 10 in. 24. Find the approximate area of a regular pentagon whose apothem measures 6 in. and each of whose sides measures approximately 8.9 in.

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  Mathematics 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.

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  Elementary Geometry for College Students

  • 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.

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  College Geometry

  A Unified Development

  • David C. Kay(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  � We confine our discussion on polygons to their existence in unified geometry. This construction also proves that a Regular Polygon can be inscribed in a given circle (its vertices lie on the circle). The converse property is also true: given a regular n -gon, a circle can be passed through its vertices, provided the sides are small enough. Consider point O in the plane and any integer n ≥ 3. Construct segment OP 1 of length less than α /2, and ray OP 2 such that m ∠ P 1 OP 2 = 360/ n ( ≤ 120) and OP 2 = OP 1 , as illustrated in Figure 4.19. Construct ray OP 3 on the oppo-site side of line OP 2 as P 1 such that m ∠ P 2 OP 3 = 360/ n and OP 3 = OP 1 . Continue constructing rays OP 4 , OP 5 , …, OP n such that for 3 ≤ i ≤ n − 2, ray OP i + 1 lies on the opposite side of line OP i as P i − 1 , m ∠ P i −1 OP i = 360/ n , and OP i +1 = OP 1 . It follows that m ∠ P n OP 1 = 360 − ( n − 1) m ∠ P 1 OP 2 = 360/ n = m ∠ P 1 OP 2 (see Problem 18, Section 2.6 ). By construction, the points P 1 , P 2 , P 3 , …, P n lie on the circle having center O and radius P 1 P 2 , and all central angles have measure 360/ n and are congruent. O P i P n P n-1 P 3 P 1 P 2 Figure 4 .19 Regular Polygons exist. To show that [ P 1 P 2 P 3 … P n P 1 ] is a convex polygon, suppose that P 3 lies on the opposite side of line P 1 P 2 as point O (Figure 4.20). Then, there exists W on line P 1 P 2 such that ( OWP 3 ) and, by construction, ray P 2 W is opposite ray P 2 P 1 . By Theorem 3 (Problem 24, Section 3.7 ), the sides of Δ OWP 2 are less than α /2. Since Δ OP 1 P 2 is an isosceles triangle having 180 College Geometry: A Unified Development legs less than α /2, ∠ OP 2 W is obtuse, and by the scalene inequality, OP 3 > OW > OP 2 = OP 1 , a contradiction. Hence P 3 lies on the O -side of line P 1 P 2 . In the same manner, P 4 lies on the O -side of line P 2 P 3 , thus on the O -side of line P 1 P 2 and so on.

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  Geometry For Dummies

  • Mark Ryan(Author)
  • 2016(Publication Date)
  • For Dummies

    (Publisher)

  equations.) Now finish with the trapezoid area formula:

  Finding the Area of Regular Polygons

  In case you’ve been dying to know how to figure the area of your ordinary, octagonal stop sign, you’ve come to the right place. (By the way, did you know that each of the eight sides of a regular-size stop sign is about 12.5 inches long? Hard to believe, but true.) In this section, you discover how to find the area of equilateral triangles, hexagons, octagons, and other shapes that have equal sides and angles.

  Presenting polygon area formulas

  A Regular Polygon is equilateral (it has equal sides) and equiangular (it has equal angles). To find the area of a Regular Polygon, you use an apothem — a segment that joins the polygon’s center to the midpoint of any side and that is perpendicular to that side ( in Figure 12-11 is an apothem).

  © John Wiley & Sons, Inc.

  FIGURE 12-11: A regular hexagon cut into six congruent, equilateral triangles.

  Area of a Regular Polygon: Use the following formula to find the area of a Regular Polygon.

  Note:

  This formula is usually written as , but if I do say so myself, the way I’ve written it, , is better. I like this way of writing it because the formula is based on the triangle area formula, : The polygon’s perimeter (p ) is related to the triangle’s base (b ), and the apothem (a ) is related to the height (h ).

  An equilateral triangle is the Regular Polygon with the fewest possible number of sides. To figure its area, you can use the Regular Polygon formula; however, it also has its own area formula (which you may remember from Chapter 7

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  Galois Theory for Beginners

  A Historical Perspective, Second Edition

  • Jörg Bewersdorff(Author)
  • 2021(Publication Date)
  • American Mathematical Society

    (Publisher)

  All the more, it seems to me that note should be taken of the discovery that in addition to those Regular Polygons, a host of others are amenable to geometric construction, for example, the heptadecagon .... Geometric constructions with straightedge and compass, generally of triangles from three given data, are a residue of classical mathematics still a part of the standard school curriculum. The significance of such exercises is less their practical application than, aside from being part of a tradition that stretches back to antiquity, to aid the student in develop-ing logical habits of thought. Construction with straightedge (unmarked ruler) and compass is limited to prescribed elementary operations that allow the construction of certain points, starting with two points sepa-rated by a distance of unit length. Thus given a set of points that have been thus constructed, the following can be additionally constructed: • Draw a circle whose midpoint is a point that has been constructed and whose radius is the distance between two constructed points. • Draw a straight line between two constructed points. • Every intersection of circles and lines drawn in the previous two steps is considered a constructed point. At first glance, there seems to be no connection between such geo-metric constructions and equations in one variable. However, as we have seen in Chapter 2, the 𝑛 th roots of unity in the complex plane, that is, the 𝑛 solutions of the equation ? 𝑛 − 1 = 0 , are the vertices of a regular 𝑛 -gon, and indeed with the unit circle as circumscribing circle. Consider Figure 7.1. If starting at the point 1 = (1, 0) we can show that the next point of the 𝑛 -gon in the counterclockwise direction, namely ? = cos( 2𝜋 𝑛 ) + 𝑖 sin( 2𝜋 𝑛 ) , can be constructed with straightedge and com-pass, then we will have succeeded in proving the regular 𝑛 -gon to be constructible.

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