DOCSLIB.ORG

A Regular Octagon Is Inscribed Inside a Square. Another Square Is Inscribed Inside the Octagon

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Regular Octagon Is Inscribed Inside a Square. Another Square Is Inscribed Inside the Octagon This brainteaser was written by Derrick Niederman. A regular octagon is inscribed inside a square. Another square is inscribed inside the octagon. What is the ratio of the area of the smaller square to the area of the larger square? © 2009 National Council of Teachers of Mathematics, Inc. Resources for Teaching Math http://illuminations.nctm.org Solution: ½. One way to see this is to rotate the blue triangles onto the top of the red trapezoids, with the hypotenuse of the triangle flush with the shorter base of the trapezoid. Then it’s pretty easy to see that if the four red and blue triangles are folded over, they’d completely cover the yellow square. In other words, the area of the yellow square is equal to the area of red trapezoids and blue triangles combined, so the ratio of the smaller square to the larger square is ½. It is also possible to calculate the area of the larger and smaller squares. Start by assuming that the length of the shorter sides of the triangles is 1 unit, as shown below. Then the hypotenuse of each triangle is 2 , and since the hypotenuse is also a side of the regular octagon, then all sides of the octagon are 2 . Consequently, the side length of the larger square is 2+ 2, and the side length of the smaller square is 1+ 2, so their 2 2 respective areas are (22+=+) 642 and (12+=+) 322. Dividing the area of the smaller square by that of the larger yields ½. If those calculations are a little too messy for you, then here is an alternative solution. Note that the length of the diagonal of the smaller square equals the distance between opposite sides of the regular octagon. However, the side length of the larger square is also equal to the distance between opposite sides of the octagon. Therefore, the side length of the larger square is 2 times the side length of the smaller square, so the ratio of the areas is ½. © 2009 National Council of Teachers of Mathematics, Inc. Resources for Teaching Math http://illuminations.nctm.org .
Load more

Recommended publications
  • Square Rectangle Triangle Diamond (Rhombus) Oval Cylinder Octagon Pentagon Cone Cube Hexagon Pyramid Sphere Star Circle
    SQUARE RECTANGLE TRIANGLE DIAMOND (RHOMBUS) OVAL CYLINDER OCTAGON PENTAGON CONE CUBE HEXAGON PYRAMID SPHERE STAR CIRCLE Powered by: www.mymathtables.com Page 1 what is Rectangle? • A rectangle is a four-sided flat shape where every angle is a right angle (90°). means "right angle" and show equal sides. what is Triangle? • A triangle is a polygon with three edges and three vertices. what is Octagon? • An octagon (eight angles) is an eight-sided polygon or eight-gon. what is Hexagon? • a hexagon is a six-sided polygon or six-gon. The total of the internal angles of any hexagon is 720°. what is Pentagon? • a plane figure with five straight sides and five angles. what is Square? • a plane figure with four equal straight sides and four right angles. • every angle is a right angle (90°) means "right ang le" show equal sides. what is Rhombus? • is a flat shape with four equal straight sides. A rhombus looks like a diamond. All sides have equal length. Opposite sides are parallel, and opposite angles are equal what is Oval? • Many distinct curves are commonly called ovals or are said to have an "oval shape". • Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. Powered by: www.mymathtables.com Page 2 What is Cube? • Six equal square faces.tweleve edges and eight vertices • the angle between two adjacent faces is ninety. what is Sphere? • no faces,sides,vertices • All points are located at the same distance from the center. what is Cylinder? • two circular faces that are congruent and parallel • faces connected by a curved surface.
    [Show full text]
  • The Other Tchaikowsky
    The Other Tchaikowsky A biographical sketch of André Tchaikowsky David A. Ferré Cover painting: André Tchaikowsky courtesy of Milein Cosman (Photograph by Ken Grundy) About the cover The portrait of André Tchaikowsky at the keyboard was painted by Milein Cosman (Mrs. Hans Keller) in 1975. André had come to her home for a visit for the first time after growing a beard. She immediately suggested a portrait be made. It was completed in two hours, in a single sitting. When viewing the finished picture, André said "I'd love to look like that, but can it possibly be me?" Contents Preface Chapter 1 - The Legacy (1935-1982) Chapter 2 - The Beginning (1935-1939) Chapter 3 - Survival (1939-1945 Chapter 4 - Years of 'Training (1945-1957) Chapter 5 - A Career of Sorts (1957-1960) Chapter 6 - Homeless in London (1960-1966) Chapter 7 - The Hampstead Years (1966-1976) Chapter 8 - The Cumnor Years (1976-1982) Chapter 9 - Quodlibet Acknowledgments List of Compositions List of Recordings i Copyright 1991 and 2008 by David A. Ferré David A. Ferré 2238 Cozy Nook Road Chewelah, WA 99109 USA [email protected] http://AndreTchaikowsky.com Preface As I maneuvered my automobile through the dense Chelsea traffic, I noticed that my passenger had become strangely silent. When I sneaked a glance I saw that his eyes had narrowed and he held his mouth slightly open, as if ready to speak but unable to bring out the words. Finally, he managed a weak, "Would you say that again?" It was April 1985, and I had just arrived in London to enjoy six months of vacation and to fulfill an overdue promise to myself.
    [Show full text]
  • Teach the Square…And the Rectangle
    f e a t u r e Teach the square…and the rectangle eaching these two geometric needs of ancient civilizations. familiar and trusted shapes, they Tshapes is like a line from an (Some believe the square symbol- represent order and stability. old song: “You can’t have one izes civilization.) Ancient peoples Because a square has four equal without the other.” needed to measure plots of land, sides, it can also be said to sym- The square and rectangle go build structures, collect taxes, bolize the seasons (spring, sum- hand in hand—maybe not like describe constellations, and create mer, fall, winter), cardinal direc- “love and marriage” or “horse calendars, all of which required tions (north, east, south, west), and carriage,” but perhaps best numbers, shapes, and calculations. and the elements (fire, water, air, understood in terms of the other. Of the basic geometric shapes in earth). Children have undoubtedly had today’s world, the rectangle is the In nature, we see no naturally experience with the two shapes in most common. The rectangle and formed squares or rectangles. objects all around them. They its square soul-mate form the Trees, branches, and leaves, for begin learning geometric shapes structure of most buildings. As example, are uneven, rounded, in a more formal way in preschool z T and are usually expected to know gae how to make and take apart N shapes by the time they finish kindergarten. by susa oto Texas Child Care has posted two ph previous articles on teaching geo- metric shapes: Go round: Teach the circle shape in Spring 2015 and Teach the triangle in Spring 2017.
    [Show full text]
  • Petrie Schemes
    Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes.
    [Show full text]
  • Rectangles with the Same Numerical Area and Perimeter
    Rectangles with the Same Numerical Area and Perimeter About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection questions, and student materials. While the primary use of Illustrations is for teacher learning about the SMP, some components may be used in the classroom with students. These include the mathematics task, student dialogue, and student materials. For additional Illustrations or to learn about a professional development curriculum centered around the use of Illustrations, please visit mathpractices.edc.org. About the Rectangles with the Same Numerical Area and Perimeter Illustration: This Illustration’s student dialogue shows the conversation among three students who are trying to find all rectangles that have the same numerical area and perimeter. After trying different rectangles of specific dimensions, students finally develop an equation that describes the relationship between the width and length of a rectangle with equal area and perimeter. Highlighted Standard(s) for Mathematical Practice (MP) MP 1: Make sense of problems and persevere in solving them. MP 2: Reason abstractly and quantitatively. MP 7: Look for and make use of structure. MP 8: Look for and express regularity in repeated reasoning. Target Grade Level: Grades 8–9 Target Content Domain: Creating Equations (Algebra Conceptual Category) Highlighted Standard(s) for Mathematical Content A.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
    [Show full text]
  • Parallelogram Rhombus Nonagon Hexagon Icosagon Tetrakaidecagon Hexakaidecagon Quadrilateral Ellipse Scalene T
    Call List parallelogram rhombus nonagon hexagon icosagon tetrakaidecagon hexakaidecagon quadrilateral ellipse scalene triangle square rectangle hendecagon pentagon dodecagon decagon trapezium / trapezoid right triangle equilateral triangle circle octagon heptagon isosceles triangle pentadecagon triskaidecagon Created using www.BingoCardPrinter.com B I N G O parallelogram tetrakaidecagon square dodecagon circle rhombus hexakaidecagon rectangle decagon octagon Free trapezium / nonagon quadrilateral heptagon Space trapezoid right isosceles hexagon hendecagon ellipse triangle triangle scalene equilateral icosagon pentagon pentadecagon triangle triangle Created using www.BingoCardPrinter.com B I N G O pentagon rectangle pentadecagon triskaidecagon hexakaidecagon equilateral scalene nonagon parallelogram circle triangle triangle isosceles Free trapezium / octagon triangle Space square trapezoid ellipse heptagon rhombus tetrakaidecagon icosagon right decagon hendecagon dodecagon hexagon triangle Created using www.BingoCardPrinter.com B I N G O right decagon triskaidecagon hendecagon dodecagon triangle trapezium / scalene pentagon square trapezoid triangle circle Free tetrakaidecagon octagon quadrilateral ellipse Space isosceles parallelogram hexagon hexakaidecagon nonagon triangle equilateral pentadecagon rectangle icosagon heptagon triangle Created using www.BingoCardPrinter.com B I N G O equilateral trapezium / pentagon pentadecagon dodecagon triangle trapezoid rectangle rhombus quadrilateral nonagon octagon isosceles Free scalene hendecagon
    [Show full text]
  • STRAND Tomorrow Only by GARDNER MACK
    ipf&ww '5ft sSEfiiwt W "T' , ";, . .! WfW THE WASHINGTON TBIES, SUNDAY, MAY 7, 1916. ,1 --L. featuring Knthcrlna Kaclrod and "Who Killed Blmon Halrd?" will be lilanche Sweet Is the featured player UcorRp Houlc Spencer. rrcsrntcd Monday, with Edna Wallace Antonio Moreno nnd Dorothy Kelly wit STARS OF THE PHOTOPLAY Charles J, ltoss, and Muriel Ho seen Monday In "Tho Supreme Temic D-2- ' This Week's Next Week's "Tho Hero it Submarine ," Is tho Cctnche fcaturod. tatlon. ' lltlo of th V. I H. E. photodramn. Mnclyn Arbucklo will be the star tklwin August nnd Orml Hawley wit foaturliu' Ch.iil ftichmen, which will Tur.Mloy In "Tho Hoform Candidate." bo thq stars Tuesday, In "Tho Socio.' Photoplays be ut Cnuiufill's Hnvoy Hin-da- y. otlmr features for the week are: HlKhwayman." Mary Johnston's fa Films son not Wednesday. Ethel Uifrrymore, In "The mous novel, "To Have and to Hold,' Waiicho Sweet will bo set-- next Kiss of Hate;" Thursday. Virginia will bo presented In fllm form Wednes- Monday In a plcturla,llon jf the l'carson, In "Hinting Loyej" Friday, day, with Mao Murray featured. CRANDALL'S. A number of attractive fenturo novel "Tho Sowers." "The Half Mil- Flank Losses, In "Tho Old Homestead;' Other attractions: Thursday. Mar Kitty Gordon, "the EngUh Lillian Dhotoolays will bo slven their first lion Drlbft." featurlnR Hamilton ncvcllo Saturday, House Peters and iiarbara Ruerlto Clark, In "Out of tho Drifts!" to- local showing weok, flnow, Is tho attraction Tennant, In "The Closed lload." Kr!rtay, Edna Wallace Hoppor, Charlei RusMlf." will be seen at CrAndall'i at Crandall's next nnd Manrriret noss, In Sunday will be scon Tucsdny.
    [Show full text]
  • Fairy Tales and Political Socialization
    FAIRY TALES AND POLITICAL SOCIALIZATION by Mehrdad Faiz Samadzadeh A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Curriculum, Teaching & Leadership University of Toronto Ontario Institute for Studies in Education © Copyright by Mehrdad Faiz Samadzadeh 2018 FAIRY TALES AND POLITICAL SOCIALIZATION Mehrdad Faiz Samadzadeh Doctor of Philosophy Department of Curriculum, Teaching & Leadership Ontario Institute for Studies in Education University of Toronto 2018 Abstract The concept of childhood is one of the many facets of modernity that entered Western consciousness in the seventeenth century. It emanated from the historical mutations of the post-Renaissance era that set in motion what Norbert Elias calls the civilizing process, one that spawned a repressive mode of socialization in tandem with the cultural and ideological hegemony of the new power elite. Accordingly, childhood became a metaphor for oppression targeting not only children, but also women, the underclass, the social outcast, and the colonized as they all were deemed “incompletely human”. From mid-nineteenth century on, however, childhood began to evince a liberating potential in tandem with the changing direction of modern Western civilization. This ushered in an alternative concept of childhood inspired by the shared characteristics between the medieval and modern child that finds expression in the works of distinguished literary figures of the Victorian era. What followed was an entire movement towards the recognition of children’s rights and status that set the context for the growing interest in childhood as a subject of historical inquiry in the twentieth century. This conceptual vicissitude of childhood is central to the present thesis which I pursue in relation to the literary genre of fairy tale.
    [Show full text]
  • Sept. 1, 2018 Aug. 30
    THE ULTIMATE LUCHA LIBRE EXPERIENCE! Aug. 30 - Sept.SePT. 1, 2018 LAS VEGAS, NEVADA Lucha Libre! Lucha Libre! The High Flying, Colorful and Character Filled World of Mexican Wrestling Is Loved By 10s Of Millions of Fans Worldwide. An ultra-exciting sport that has transcended time; from the days the ancient Aztec warriors donned masks to go to war, to today’s mainstream pop culture scene. And, while Mexico City may be the center of the lucha libre universe, from Southern California to South America, London to Qatar, and all parts in between, lucha libre events are drawing crowds larger than any other non-WWE form of pro wrestling. Mexico’s Expo: A Huge Success! Fans Globally Now Demand It! For a number of years, Mexico City's Lucha Libre La Experencia was the premiere destination for lucha libre fans from across the globe and local families who wanted an experience like no other. Thousands of fans & families would make the pilgrimage to Mexico City to experience 3 days of all things lucha libre. La Experencia was the only place to see events from the top lucha libre companies, and with lucha libre's biggest stars all under one roof. Featuring multiple events, plus the chance to meet and interact with their heroes live and in person, there was nothing like the Expo! EXPO-LUCHA! Now, Masked Republic, the only company in the world “Dedicated to Expanding Lucha Libre Beyond The Borders of Mexico”™ , brings the first ever EXPO-LUCHA to the United States... Right to the expo capital of the world, the premiere family and adult destination… LAS VEGAS! EXPO-LUCHA will feature over 100 of the world's biggest lucha libre icons, legends, and superstars, - with more than 25 million combined social media followers.
    [Show full text]
  • Rhombus Rectangle Square Trapezoid Kite NOTES
    Geometry Notes G.9 Rhombus, Rectangle, Square, Trapezoid, Kite Mrs. Grieser Name: _________________________________________ Date: _________________ Block: _______ Rhombuses, Rectangles, Squares The Venn diagram below describes the relationship between different kinds of parallelograms: A rhombus is a parallelogram with four congruent sides A rectangle is a parallelogram with four right angles A square is a parallelogram with four congruent sides and four right angles Corollaries: A quadrilateral is a rhombus IFF it has four congruent sides A quadrilateral is a rectangles IFF it has four right angles A quadrilateral is a square IFF it is a rhombus and a rectangle. Since rhombuses, squares, and rectangles are parallelograms, they have all the properties of parallelograms (opposite sides parallel, opposite angles congruent, diagonals bisect each other, etc.) In addition… Rhombus Rectangle Square 4 congruent sides 4 right angles 4 congruent sides diagonals bisect angles diagonals congruent diagonals bisect each other diagonals perpendicular diagonals perpendicular 4 right angles diagonals congruent Theorems: A parallelogram is a rhombus IFF its diagonals are perpendicular. A parallelogram is a rhombus IFF each diagonal bisects a pair of opposite angles. A parallelogram is a rectangle IFF its diagonals are congruent. Examples: 1) Given rhombus DEFG, are the statements 2) Classify the parallelogram and find missing sometimes, always, or never true: values: a) b) a) D F b) D E c) DG GF 3) Given rhombus WXYZ 4) Given rectangle PQRS and and mXZY 34, find: mRPS 62 and QS=18, a) mWZV b) WY c) XY find: a) mQPR b) mPTQ c) ST Geometry Notes G.9 Rhombus, Rectangle, Square, Trapezoid, Kite Mrs.
    [Show full text]
  • Two-Dimensional Figures a Plane Is a Flat Surface That Extends Infinitely in All Directions
    NAME CLASS DATE Two-Dimensional Figures A plane is a flat surface that extends infinitely in all directions. A parallelogram like the one below is often used to model a plane, but remember that a plane—unlike a parallelogram—has no boundaries or sides. A plane figure or two-dimensional figure is a figure that lies completely in one plane. When you draw, either by hand or with a computer program, you draw two-dimensional figures. Blueprints are two-dimensional models of real-life objects. Polygons are closed, two-dimensional figures formed by three or more line segments that intersect only at their endpoints. These figures are polygons. These figures are not polygons. This is not a polygon A heart is not a polygon A circle is not a polygon because it is an open because it is has curves. because it is made of figure. a curve. Polygons are named by the number of sides and angles they have. A polygon always has the same number of sides as angles. Listed on the next page are the most common polygons. Each of the polygons shown is a regular polygon. All the angles of a regular polygon have the same measure and all the sides are the same length. SpringBoard® Course 1 Math Skills Workshop 89 Unit 5 • Getting Ready Practice MSW_C1_SE.indb 89 20/07/19 1:05 PM Two-Dimensional Figures (continued) Triangle Quadrilateral Pentagon Hexagon 3 sides; 3 angles 4 sides; 4 angles 5 sides; 5 angles 6 sides; 6 angles Heptagon Octagon Nonagon Decagon 7 sides; 7 angles 8 sides; 8 angles 9 sides; 9 angles 10 sides; 10 angles EXAMPLE A Classify the polygon.
    [Show full text]
  • Patterned Triply Periodic Polyhedra
    Patterned Triply Periodic Polyhedra Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-3036, USA E-mail: [email protected] Web Site: http://www.d.umn.edu/˜ddunham/ Abstract This paper discusses repeating patterns on infinite skew polyhedra, which are triply periodic polyhedra. We exhibit patterns on each of the three regular skew polyhedra. These patterns are each related to corresponding repeating patterns in the hyperbolic plane. This correspondence will be explained in the paper. 1. Introduction A number of people, including M.C. Escher, created convex polyhedra with patterns on them. Later, in 1977 Doris Schattschneider and Wallace Walker designed non-convex rings of polyhedra, called Kaleidocycles, that could be rotated, which are described in [Sch05]. The goal of this paper is to start an investigation of repeating patterns on infinite skew polyhedra — i.e. triply periodic polyhedra. Figure 1 shows a finite piece of such a pattern. Figure 1: A pattern of fish on the tessellation {6, 6|3}. We begin with a discussion of infinite skew polyhedra and show how they are related to tessellations of the hyperbolic plane. This relationship can also be applied to repeating patterns on those respective surfaces. Then we present patterns on each of the three regular triply periodic polyhedra. Finally, we indicate possible directions of further investigation. 2. Patterns, Hyperbolic Geometry, and Infinite Skew Polyhedra. A repeating pattern is a pattern made up of congruent copies of a basic subpattern or motif. There can be repeating patterns on the Euclidean plane, hyperbolic plane, sphere, and polyhedra.
    [Show full text]