Build a Tetrahedral Kite
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THE GEOMETRY of PYRAMIDS One of the More Interesting Solid
THE GEOMETRY OF PYRAMIDS One of the more interesting solid structures which has fascinated individuals for thousands of years going all the way back to the ancient Egyptians is the pyramid. It is a structure in which one takes a closed curve in the x-y plane and connects straight lines between every point on this curve and a fixed point P above the centroid of the curve. Classical pyramids such as the structures at Giza have square bases and lateral sides close in form to equilateral triangles. When the closed curve becomes a circle one obtains a cone and this cone becomes a cylindrical rod when point P is moved to infinity. It is our purpose here to discuss the properties of all N sided pyramids including their volume and surface area using only elementary calculus and geometry. Our starting point will be the following sketch- The base represents a regular N sided polygon with side length ‘a’ . The angle between neighboring radial lines r (shown in red) connecting the polygon vertices with its centroid is θ=2π/N. From this it follows, by the law of cosines, that the length r=a/sqrt[2(1- cos(θ))] . The area of the iscosolis triangle of sides r-a-r is- a a 2 a 2 1 cos( ) A r 2 T 2 4 4 (1 cos( ) From this we have that the area of the N sided polygon and hence the pyramid base will be- 2 2 1 cos( ) Na A N base 2 4 1 cos( ) N 2 It readily follows from this result that a square base N=4 has area Abase=a and a hexagon 2 base N=6 yields Abase= 3sqrt(3)a /2. -
Math 366 Lecture Notes Section 11.4 – Geometry in Three Dimensions
Section 11-4 Math 366 Lecture Notes Section 11.4 – Geometry in Three Dimensions Simple Closed Surfaces A simple closed surface has exactly one interior, no holes, and is hollow. A sphere is the set of all points at a given distance from a given point, the center . A sphere is a simple closed surface. A solid is a simple closed surface with all interior points. (see p. 726) A polyhedron is a simple closed surface made up of polygonal regions, or faces . The vertices of the polygonal regions are the vertices of the polyhedron, and the sides of each polygonal region are the edges of the polyhedron. (see p. 726-727) A prism is a polyhedron in which two congruent faces lie in parallel planes and the other faces are bounded by parallelograms. The parallel faces of a prism are the bases of the prism. A prism is usually names after its bases. The faces other than the bases are the lateral faces of a prism. A right prism is one in which the lateral faces are all bounded by rectangles. An oblique prism is one in which some of the lateral faces are not bounded by rectangles. To draw a prism: 1) Draw one of the bases. 2) Draw vertical segments of equal length from each vertex. 3) Connect the bottom endpoints to form the second base. Use dashed segments for edges that cannot be seen. A pyramid is a polyhedron determined by a polygon and a point not in the plane of the polygon. The pyramid consists of the triangular regions determined by the point and each pair of consecutive vertices of the polygon and the polygonal region determined by the polygon. -
VOLUME of POLYHEDRA USING a TETRAHEDRON BREAKUP We
VOLUME OF POLYHEDRA USING A TETRAHEDRON BREAKUP We have shown in an earlier note that any two dimensional polygon of N sides may be broken up into N-2 triangles T by drawing N-3 lines L connecting every second vertex. Thus the irregular pentagon shown has N=5,T=3, and L=2- With this information, one is at once led to the question-“ How can the volume of any polyhedron in 3D be determined using a set of smaller 3D volume elements”. These smaller 3D eelements are likely to be tetrahedra . This leads one to the conjecture that – A polyhedron with more four faces can have its volume represented by the sum of a certain number of sub-tetrahedra. The volume of any tetrahedron is given by the scalar triple product |V1xV2∙V3|/6, where the three Vs are vector representations of the three edges of the tetrahedron emanating from the same vertex. Here is a picture of one of these tetrahedra- Note that the base area of such a tetrahedron is given by |V1xV2]/2. When this area is multiplied by 1/3 of the height related to the third vector one finds the volume of any tetrahedron given by- x1 y1 z1 (V1xV2 ) V3 Abs Vol = x y z 6 6 2 2 2 x3 y3 z3 , where x,y, and z are the vector components. The next question which arises is how many tetrahedra are required to completely fill a polyhedron? We can arrive at an answer by looking at several different examples. Starting with one of the simplest examples consider the double-tetrahedron shown- It is clear that the entire volume can be generated by two equal volume tetrahedra whose vertexes are placed at [0,0,sqrt(2/3)] and [0,0,-sqrt(2/3)]. -
The Mars Pentad Time Pyramids the Quantum Space Time Fractal Harmonic Codex the Pentagonal Pyramid
The Mars Pentad Time Pyramids The Quantum Space Time Fractal Harmonic Codex The Pentagonal Pyramid Abstract: Early in this author’s labors while attempting to create the original Mars Pentad Time Pyramids document and pyramid drawings, a failure was experienced trying to develop a pentagonal pyramid with some form of tetrahedral geometry. This pentagonal pyramid is now approached again and refined to this tetrahedral criteria, creating tetrahedral angles in the pentagonal pyramid, and pentagonal [54] degree angles in the pentagon base for the pyramid. In the process another fine pentagonal pyramid was developed with pure pentagonal geometries using the value for ancient Egyptian Pyramid Pi = [22 / 7] = [aPi]. Also used is standard modern Pi and modern Phi in one of the two pyramids shown. Introduction: Achieved are two Pentagonal Pyramids: One creates tetrahedral angle [54 .735~] in the Side Angle {not the Side Face Angle}, using a novel height of the value of tetrahedral angle [19 .47122061] / by [5], and then the reverse tetrahedral angle is accomplished with the same tetrahedral [19 .47122061] angle value as the height, but “Harmonic Codexed” to [1 .947122061]! This achievement of using the second height mentioned, proves aspects of the Quantum Space Time Fractal Harmonic Codex. Also used is Height = [2], which replicates the [36] and [54] degree angles in the pentagon base to the Side Angles of the Pentagonal Pyramid. I have come to understand that there is not a “perfect” pentagonal pyramid. No matter what mathematical constants or geometry values used, there will be a slight factor of error inherent in the designs trying to attain tetrahedra. -
Pentagonal Pyramid
Chapter 9 Surfaces and Solids Copyright © Cengage Learning. All rights reserved. Pyramids, Area, and 9.2 Volume Copyright © Cengage Learning. All rights reserved. Pyramids, Area, and Volume The solids (space figures) shown in Figure 9.14 below are pyramids. In Figure 9.14(a), point A is noncoplanar with square base BCDE. In Figure 9.14(b), F is noncoplanar with its base, GHJ. (a) (b) Figure 9.14 3 Pyramids, Area, and Volume In each space pyramid, the noncoplanar point is joined to each vertex as well as each point of the base. A solid pyramid results when the noncoplanar point is joined both to points on the polygon as well as to points in its interior. Point A is known as the vertex or apex of the square pyramid; likewise, point F is the vertex or apex of the triangular pyramid. The pyramid of Figure 9.14(b) has four triangular faces; for this reason, it is called a tetrahedron. 4 Pyramids, Area, and Volume The pyramid in Figure 9.15 is a pentagonal pyramid. It has vertex K, pentagon LMNPQ for its base, and lateral edges and Although K is called the vertex of the pyramid, there are actually six vertices: K, L, M, N, P, and Q. Figure 9.15 The sides of the base and are base edges. 5 Pyramids, Area, and Volume All lateral faces of a pyramid are triangles; KLM is one of the five lateral faces of the pentagonal pyramid. Including base LMNPQ, this pyramid has a total of six faces. The altitude of the pyramid, of length h, is the line segment from the vertex K perpendicular to the plane of the base. -
Unit 6 Visualising Solid Shapes(Final)
• 3D shapes/objects are those which do not lie completely in a plane. • 3D objects have different views from different positions. • A solid is a polyhedron if it is made up of only polygonal faces, the faces meet at edges which are line segments and the edges meet at a point called vertex. • Euler’s formula for any polyhedron is, F + V – E = 2 Where F stands for number of faces, V for number of vertices and E for number of edges. • Types of polyhedrons: (a) Convex polyhedron A convex polyhedron is one in which all faces make it convex. e.g. (1) (2) (3) (4) 12/04/18 (1) and (2) are convex polyhedrons whereas (3) and (4) are non convex polyhedron. (b) Regular polyhedra or platonic solids: A polyhedron is regular if its faces are congruent regular polygons and the same number of faces meet at each vertex. For example, a cube is a platonic solid because all six of its faces are congruent squares. There are five such solids– tetrahedron, cube, octahedron, dodecahedron and icosahedron. e.g. • A prism is a polyhedron whose bottom and top faces (known as bases) are congruent polygons and faces known as lateral faces are parallelograms (when the side faces are rectangles, the shape is known as right prism). • A pyramid is a polyhedron whose base is a polygon and lateral faces are triangles. • A map depicts the location of a particular object/place in relation to other objects/places. The front, top and side of a figure are shown. Use centimetre cubes to build the figure. -
Performance Based Learning and Assessment Task Kite Project
Performance Based Learning and Assessment Task Kite Project I. ASSESSSMENT TASK OVERVIEW & PURPOSE: The students are instructed to: research the history, science, and design of kites; design a blueprint of their kite and find the measurements, scale factor, area, and perimeter of their blueprint; and construct and fly their kite. II. UNIT AUTHOR: Leslie Hindman, Washington-Lee High School, Arlington Public Schools III. COURSE: Geometry IV. CONTENT STRAND: Geometry, Measurement V. OBJECTIVES: Students will be able to • Design a scale model of a kite • Measure the dimensions and angles of the scale model • Determine the scale factor of the scale model • Compute the area and perimeter of the scale model • Construct and fly a kite VI. REFERENCE/RESOURCE MATERIALS: For Research: computer access For Scale Model Drawing: ruler, protractor, graph paper, calculator, Geometry SOL formula sheet For Kites: tissue paper, plastic table cloths, small wooden dowels, straws, yarn, fishing wire, markers, scissors, tape, glue VII. PRIMARY ASSESSMENT STRATEGIES: The task includes an assessment component that performs two functions: (1) for the student it will be a checklist and provide a self-assessment and (2) for the teacher it will be used as a rubric. The attached assessment list will assess the research section, scale model drawing, and kite construction. VIII. EVALUATION CRITERIA: Assessment List, corresponding rubric. IX. INSTRUCTIONAL TIME: 2 ninety minute class periods for research, scale model drawings, and kite construction. 1/2 ninety minute class period for flying kites. Kite Project Strand Geometry, Measurement Mathematical Objective(s) Students will be able to: • Design a scale model of a kite • Find the measures of the sides and angles of the scale model • Determine the scale factor of the scale model • Compute the area and perimeter of the scale model Related SOL • G.12 The student will make a model of a three-dimensional figure from a two- dimensional drawing and make a two-dimensional representation of a three- dimensional object. -
Classroom Capsules
CLASSROOM CAPSULES EDITOR Michael K. Kinyon Indiana University South Bend South Bend, IN 46634 Classroom Capsules consists primarily of short notes (1–3 pages) that convey new mathematical in- sights and effective teaching strategies for college mathematics instruction. Please submit manuscripts prepared according to the guidelines on the inside front cover to the Editor, Michael K. Kinyon, Indiana University South Bend, South Bend, IN 46634. The Theorem of Cosines for Pyramids Alexander Kheyfits (alexander.kheyfi[email protected]), Bronx Community College, CUNY, Bronx, NY 10453 The classical three-millennia-old Pythagorean theorem is still irresistibly attractive for lovers of mathematics—see, for example, [1] or a recent survey [8] for its numer- ous old and new generalizations. It can be extended to 3-space in various ways. A straightforward extension states that the square of a diagonal of a right rectangular parallelepiped is equal to the sum of squares of its adjacent edges. A less trivial gener- alization is the following result, which can be found in good collections of geometric problems as well as in a popular calculus textbook [7, p. 858]: (PTT) The Pythagorean theorem for tetrahedra with a trirectangular ver- tex. In a tetrahedron with a trirectangular vertex, the square of the area of the op- posite face to this vertex is equal to the sum of the squares of the areas of three other faces. Consider a triangle with sides a, b, c and the angle C opposite side c. The classical Pythagorean theorem is a particular case of the Theorem or Law of Cosines, c2 = a2 + b2 − 2ab cos C, if we specify here C to be a right angle. -
Putting the Icosahedron Into the Octahedron
Forum Geometricorum Volume 17 (2017) 63–71. FORUM GEOM ISSN 1534-1178 Putting the Icosahedron into the Octahedron Paris Pamfilos Abstract. We compute the dimensions of a regular tetragonal pyramid, which allows a cut by a plane along a regular pentagon. In addition, we relate this construction to a simple construction of the icosahedron and make a conjecture on the impossibility to generalize such sections of regular pyramids. 1. Pentagonal sections The present discussion could be entitled Organizing calculations with Menelaos, and originates from a problem from the book of Sharygin [2, p. 147], in which it is required to construct a pentagonal section of a regular pyramid with quadrangular F J I D T L C K H E M a x A G B U V Figure 1. Pentagonal section of quadrangular pyramid basis. The basis of the pyramid is a square of side-length a and the pyramid is assumed to be regular, i.e. its apex F is located on the orthogonal at the center E of the square at a distance h = |EF| from it (See Figure 1). The exercise asks for the determination of the height h if we know that there is a section GHIJK of the pyramid by a plane which is a regular pentagon. The section is tacitly assumed to be symmetric with respect to the diagonal symmetry plane AF C of the pyramid and defined by a plane through the three points K, G, I. The first two, K and G, lying symmetric with respect to the diagonal AC of the square. -
INTRODUCTION to QUADRILATERALS Square Rectangle Rhombus Parallelogram
INTRODUCTION TO QUADRILATERALS square rectangle rhombus parallelogram trapezoid kite Mathematicians define regular quadrilaterals according to their attributes. a) Quadrilaterals are two-dimensional closed figures that have four sides and four angles. b) The sum of their four angles is 360 degrees. c) When we trace diagonals from one opposite corner to another in a quadrilateral, these lines always intersect (pass it through) and in some cases they bisect (cut the other line in half). d) Regular quadrilaterals are named square, rectangle, parallelogram, rhombus, trapezoid, and kite. e) VERY IMPORTANT TO REMEMBER THIS! : Although these are the names of the quadrilaterals, these words are also the names of groups of quadrilaterals. Each group has its own characteristics or attributes that distinguish them from the others. For instance, the figure square belongs obviously to the group of squares, but also belongs to the groups of trapezoids, parallelograms, rectangles, rhombuses, and kites, because it shares similar attributes with them. However, the group of squares has only one figure: square. This is because no other quadrilateral shares the attributes of the squares, which are having four equal sides and four right angles. TRAPEZOIDS a) Trapezoids form the largest group of quadrilaterals. They include the figures of square, rectangle, parallelogram, rhombus, and of course, trapezoid. (the figure kite does not belong to the group of trapezoids) b) Trapezoids have only one attribute: they have at least one parallel line. Also, a. They may or may not have two or even four equal in length sides. b. They may or may not have opposite sides of the same length c. -
2D and 3D Shapes.Pdf
& • Anchor 2 - D Charts • Flash Cards 3 - D • Shape Books • Practice Pages rd Shapesfor 3 Grade by Carrie Lutz T hank you for purchasing!!! Check out my store: http://www.teacherspayteachers.com/Store/Carrie-Lutz-6 Follow me for notifications of freebies, sales and new arrivals! Visit my BLOG for more Free Stuff! Read My Blog Post about Teaching 3 Dimensional Figures Correctly Credits: Carrie Lutz©2016 2D Shape Bank 3D Shape Bank 3 Sided 5 Sided Prisms triangular prism cube rectangular prism triangle pentagon 4 Sided rectangle square pentagonal prism hexagonal prism octagonal prism Pyramids rhombus trapezoid 6 Sided 8 Sided rectangular square triangular pyramid pyramid pyramid Carrie LutzCarrie CarrieLutz pentagonal hexagonal © hexagon octagon © 2016 pyramid pyramid 2016 Curved Shapes CURVED SOLIDS oval circle sphere cone cylinder Carrie Lutz©2016 Carrie Lutz©2016 Name _____________________ Side Sort Date _____________________ Cut out the shapes below and glue them in the correct column. More than 4 Less than 4 Exactly 4 Carrie Lutz©2016 Name _____________________ Name the Shapes Date _____________________ 1. Name the Shape. 2. Name the Shape. 3. Name the Shape. ____________________________________ ____________________________________ ____________________________________ 4. Name the Shape. 5. Name the Shape. 6. Name the Shape. ____________________________________ ____________________________________ ____________________________________ 4. Name the Shape. 5. Name the Shape. 6. Name the Shape. ____________________________________ ____________________________________ ____________________________________ octagon circle square rhombus triangle hexagon pentagon rectangle trapezoid Carrie Lutz©2016 Faces, Edges, Vertices Name _____________________ and Date _____________________ 1. Name the Shape. 2. Name the Shape. 3. Name the Shape. ____________________________________ ____________________________________ ____________________________________ _____ faces _____ faces _____ faces _____Edges _____Edges _____Edges _____Vertices _____Vertices _____Vertices 4. -
When Is a Tangential Quadrilateral a Kite?
Forum Geometricorum b Volume 11 (2011) 165–174. b b FORUM GEOM ISSN 1534-1178 When is a Tangential Quadrilateral a Kite? Martin Josefsson Abstract. We prove 13 necessary and sufficient conditions for a tangential quadri- lateral to be a kite. 1. Introduction A tangential quadrilateral is a quadrilateral that has an incircle. A convex quadrilateral with the sides a, b, c, d is tangential if and only if a + c = b + d (1) according to the Pitot theorem [1, pp.65–67]. A kite is a quadrilateral that has two pairs of congruent adjacent sides. Thus all kites has an incircle since its sides satisfy (1). The question we will answer here concerns the converse, that is, what additional property a tangential quadrilateral must have to be a kite? We shall prove 13 such conditions. To prove two of them we will use a formula for the area of a tangential quadrilateral that is not so well known, so we prove it here first. It is given as a problem in [4, p.29]. Theorem 1. A tangential quadrilateral with sides a, b, c, d and diagonals p,q has the area 1 2 2 K = 2 (pq) − (ac − bd) . p Proof. A convex quadrilateral with sides a, b, c, d and diagonals p,q has the area 1 2 2 2 2 2 2 2 K = 4 4p q − (a − b + c − d ) (2) p according to [6] and [14]. Squaring the Pitot theorem (1) yields 2 2 2 2 a + c + 2ac = b + d + 2bd. (3) Using this in (2), we get 1 2 2 K = 4 4(pq) − (2bd − 2ac) p and the formula follows.