Perimeter and Area
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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. -
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. -
If Two Polygons Are Similar, Then Their Areas Are Proportional to the Square of the Scale Factor SOLUTION: Between Them
Chapter 10 Study Guide and Review State whether each sentence is true or false. If 6. The measure of each radial angle of a regular n-gon false, replace the underlined term to make a is . true sentence. 1. The center of a trapezoid is the perpendicular SOLUTION: distance between the bases. false; central SOLUTION: ANSWER: false; height false; central ANSWER: 7. The apothem of a polygon is the perpendicular false; height distance between any two parallel bases. 2. A slice of pizza is a sector of a circle. SOLUTION: false; height of a parallelogram SOLUTION: true ANSWER: false; height of a parallelogram ANSWER: true 8. The height of a triangle is the length of an altitude drawn to a given base. 3. The center of a regular polygon is the distance from the middle to the circle circumscribed around the SOLUTION: polygon. true SOLUTION: ANSWER: false; radius true ANSWER: 9. Explain how the areas of two similar polygons are false; radius related. 4. The segment from the center of a square to the SOLUTION: corner can be called the radius of the square. If two polygons are similar, then their areas are proportional to the square of the scale factor SOLUTION: between them. true ANSWER: ANSWER: If two polygons are similar, then their areas are true proportional to the square of the scale factor 5. A segment drawn perpendicular to a side of a regular between them. polygon is called an apothem of the polygon. SOLUTION: true ANSWER: true eSolutions Manual - Powered by Cognero Page 1 Chapter 10 Study Guide and Review 10. -
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. -
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. -
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. -
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. -
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 -
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. -
Areas of Regular Polygons Finding the Area of an Equilateral Triangle
Areas of Regular Polygons Finding the area of an equilateral triangle The area of any triangle with base length b and height h is given by A = ½bh The following formula for equilateral triangles; however, uses ONLY the side length. Area of an equilateral triangle • The area of an equilateral triangle is one fourth the square of the length of the side times 3 s s A = ¼ s2 s A = ¼ s2 Finding the area of an Equilateral Triangle • Find the area of an equilateral triangle with 8 inch sides. Finding the area of an Equilateral Triangle • Find the area of an equilateral triangle with 8 inch sides. 2 A = ¼ s Area3 of an equilateral Triangle A = ¼ 82 Substitute values. A = ¼ • 64 Simplify. A = • 16 Multiply ¼ times 64. A = 16 Simplify. Using a calculator, the area is about 27.7 square inches. • The apothem is the F height of a triangle A between the center and two consecutive vertices H of the polygon. a E G B • As in the activity, you can find the area o any regular n-gon by dividing D C the polygon into congruent triangles. Hexagon ABCDEF with center G, radius GA, and apothem GH A = Area of 1 triangle • # of triangles F A = ( ½ • apothem • side length s) • # H of sides a G B = ½ • apothem • # of sides • side length s E = ½ • apothem • perimeter of a D C polygon Hexagon ABCDEF with center G, This approach can be used to find radius GA, and the area of any regular polygon. apothem GH Theorem: Area of a Regular Polygon • The area of a regular n-gon with side lengths (s) is half the product of the apothem (a) and the perimeter (P), so The number of congruent triangles formed will be A = ½ aP, or A = ½ a • ns. -
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. -
Influence of Coal Dust on Premixed Turbulent Methane-Air Flames Scott
Influence of Coal Dust on Premixed Turbulent Methane-Air Flames Scott R. Rockwell A Dissertation Submitted to the Faculty of Worcester Polytechnic Institute In partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Fire Protection Engineering July 2012 APPROVED: _____________________________________________ Professor Ali S. Rangwala, Advisor _____________________________________________ Professor Kathy A. Notarianni, Head of Department _____________________________________________ Professor Simon W. Evans _____________________________________________ Professor Sanjeeva Balasuriya _____________________________________________ Dr. Alfonso F. Ibarreta _____________________________________________ Professor Forman A. Williams Table of Contents Table of Contents ............................................................................................................................ 2 List of figures .................................................................................................................................. 5 List of tables .................................................................................................................................... 6 Nomenclature .................................................................................................................................. 7 Acknowledgments........................................................................................................................... 8 Abstract ..........................................................................................................................................