1 SOLIDS, NETS, AND CROSS SECTIONS Polyhedra In this section, we will examine various three-dimensional figures, known as solids. We begin with a discussion of polyhedra. Polyhedron A polyhedron is a three-dimensional solid with the following properties: 1. A polyhedron is composed entirely of polygons; each of these polygons is known as a face. 2. The segment where two polygons intersect is known as an edge, and the edge is a shared side of the two polygons. 3. The point where three or more edges intersect is known as a vertex. The vertices of the polyhedron are the vertices of the polygons. The plural of polyhedron is polyhedra. Polyhedra are named according to the number of their faces, as found in the table below. (A very brief and incomplete listing is found here.) Number of Faces Name of Polyhedron 4 Tetrahedron 5 Pentahedron 6 Hexahedron 7 Heptahedron 8 Octahedron 9 Enneahedron 10 Decahedron 11 Hendecahedron 12 Dodecahedron 14 Tetradecahedron 20 Icosahedron 24 Icositetrahedron 32 Icosidodecahedron n n-hedron Notice that this naming system is very general, as it only counts the number of faces -- without any regard to the types of polygons that comprise the polyhedron.
2 Examples List the faces, edges, and vertices for each of the polyhedra shown below. Then name the polyhedron according to the table above. F G 1. Solution: The polyhedron has six faces: ABCD, EFGH, BFGC, AEHD, ABFE, and DCGH. B E C H The polyhedron has twelve edges: AB, BC, CD, AD, EF, FG, GH, EH, BF, CG, AE, and DH. A D The polyhedron has eight vertices: A, B, C, D, E, F, G, and H. Since the polyhedron has six faces, it is called a hexahedron. I 2. Solution: The polyhedron has six faces: IJK, IKL, ILM, INM, IJN, and JKLMN. K L M The polyhedron has ten edges: IJ, IK, IL, IM, IN, JK, KL, LM, MN, and JN. The polyhedron has six vertices: I, J, K, L, M, and N. J N Since the polyhedron has six faces, it is called a hexahedron. Even though the two solids in the examples above are very different from each other, notice that each one can be classified as a hexahedron. In the next section, we will begin to learn about more specific categories of polyhedra. Exercises 1. Draw a tetrahedron and label its vertices. Then list all of its faces and edges. 2. Draw a pentahedron and label its vertices. Then list all of its faces and edges.
3 Prisms Prism A prism is a polyhedron with the following properties: 1. Two congruent polygonal faces lie in parallel planes; each of these two faces is known as a base. 2. A segment known as a lateral edge joins each vertex of a base with its corresponding vertex on the other base. 3. The remaining faces are parallelograms, and are known as lateral faces. If the lateral edges are perpendicular to the bases, the lateral faces are rectangles and the prism is a right prism. Otherwise, the prism is an oblique prism. A prism is named by the shape of its base. If its base is regular (i.e. all sides congruent and all angles congruent), then it is called a regular prism. To draw a prism using transformations: Draw a base and then translate it onto a plane other than itself. (This image is the other base.) Then connect the corresponding vertices of both bases to form the lateral edges of the prism. Examples Name each of the prisms below, and then classify it as right or oblique. 1. Solution: This is a pentagonal prism, since the bases are pentagons. It is a right prism, since the lateral edges are perpendicular to the bases. 2. Solution: This is a triangular prism, since the bases are triangles. (Notice that the bases need only to be congruent and to be in parallel planes;
4 they don t need to be at the top and bottom of the diagram.) It is right prism, since the lateral edges are perpendicular to the bases. 3. Solution: This is a regular hexagonal prism, since the bases are regular hexagons. It is an oblique prism, since the lateral edges are not perpendicular to the bases. Note: The bases are regular hexagons. 4. Solution: This is a rectangular prism, since the bases are rectangles. It is a right prism, since the lateral edges are perpendicular to the bases. Exercises 1. Draw a right octagonal prism. Then state the number of faces (including the bases), edges, lateral faces, lateral edges, and vertices. 2. Draw an oblique regular triangular prism. Then state the number of faces (including the bases), edges, lateral faces, lateral edges, and vertices. 3. Build the following prisms with Polydrons (or a similar manipulative where various polygons can be interlocked together to form solids): a. A triangular prism b. A cube c. A pentagonal prism
5 Pyramids Pyramid A pyramid is a polyhedron with the following properties: 1. One polygon represents the base. 2. A point, known as the apex, lies on a plane other then the plane of the base itself. 3. A segment known as a lateral edge joins each vertex of the base with the apex. 4. The remaining faces are triangles and are known as lateral faces. A pyramid is named by the shape of its base. If its base is regular and its lateral edges are congruent, then it is called a regular pyramid. Examples Name each of the pyramids below. 1. Solution: This is a triangular pyramid. 2. Solution: This is a square pyramid. Note: The base is a square. 3. Solution: This is a heptagonal pyramid.
6 Exercises 1. Draw a pentagonal pyramid. Then state the number of faces (including the base), edges, lateral faces, lateral edges, and vertices. 2. Draw a hexagonal pyramid. Then state the number of faces (including the base), edges, lateral faces, lateral edges, and vertices. 3. Build the following pyramids with Polydrons (or a similar manipulative where various polygons can be interlocked together to form solids): a. Triangular pyramid b. Square pyramid c. Pentagonal pyramid Cylinders A cylinder is very similar to a prism, in that its two bases are congruent. The bases, however, are circles instead of polygons. Because of the curved shape of the base, there are no lateral faces, but instead one smooth lateral surface. (If you think of a cylinder like a soup can, then the lateral surface would be represented by the label.) The segment joining the centers of the two bases is called the axis of the cylinder. If the axis is perpendicular to the bases, then the cylinder is a right cylinder. If the axis is not perpendicular to the bases, then the cylinder is an oblique cylinder. Diagrams of cylinders: Right Cylinder Oblique Cylinder Even though a cylinder is a three-dimensional solid, it cannot be classified as a polyhedron because it is not made up of polygons. (A circle is not a polygon since it is curved rather than composed of segments.)
7 Cones A cone is very similar to a pyramid. Its base, however, is a circle rather than a polygon. It has an apex just as a pyramid has an apex, but due to the curved shape of the base, there are no lateral faces, but instead a smooth lateral surface. (Since a cone is not composed of polygons, it cannot be classified as a polyhedron.) The segment joining the apex to the center of the base is called the axis of the cone. If the axis is perpendicular to the base, then the cone is a right cone. If the axis is not perpendicular to the base, then the cylinder is an oblique cone. Diagrams of cones: Right Cone Oblique Cone Platonic Solids There are five special polyhedra which are known as the Platonic Solids. Each of the Platonic Solids is made up of congruent regular polygons. The five Platonic Solids are listed and illustrated below: 1. The Tetrahedron The tetrahedron is made up of four congruent equilateral triangles. 2. The Cube The cube is made up of six congruent squares.
8 3. The Octahedron The octahedron is made up of eight congruent equilateral triangles. 4. The Dodecahedron The dodecahedron is made up of twelve congruent regular pentagons. 5. The Icosahedron The icosahedron is made up of twenty congruent equilateral triangles. Exercise Build the five Platonic Solids with Polydrons (or a similar manipulative where various polygons can be interlocked together to form solids).
9 Nets Suppose that we wanted to use a pattern to create a solid in the same way that a dressmaker uses a pattern to make a dress. Such a pattern is called a net. Net A net is a two-dimensional pattern for a solid. Examples Imagine being able to unfold each of the solids below, and draw a possible net for each solid. Assume that all bases are regular polygons. (If a manipulative such as Polydrons is available, build these solids and then unfold them to see what each net looks like.)
10 4. Solutions: The original solid is shown at the left, with a possible net drawn on the right. Solid Net 1. 2.
11 Solutions (continued): The original solid is shown at the left, with a possible net drawn on the right. Solid Net 3. 4.
12 Exercises Answer the following. 1. Draw a regular right hexagonal pyramid and then draw its net. 2. Draw a rectangular prism and then draw its net. 3. Draw a regular right octagonal prism and then draw its net. 4. Draw a net for the following oblique hexagonal prism. (Hint: In an oblique prism, what are the lateral faces made of?) Note: The bases are regular hexagons. 5. The two nets below represent different nets for the same cube. There are eleven different nets (i.e. nine others) for a cube. Draw as many as you can. (Note: Do not count congruent nets twice; if a net can be rotated or reflected and it looks identical to another one, then the nets are congruent.) 6. Draw a net for a right cylinder. (In your answer, the bases will barely be connected to the rest of the net, but this still shows what a cylinder looks like when it is made into a two-dimensional pattern.) 7. Draw a net for a right cone. (As with the cylinder above, the base will barely be connected to the rest of the net. This one is difficult; you may want to experiment with a piece of paper and scissors to try to create the lateral surface of a cone.)
13 Cross Sections We will now turn our attention to the cross sections of solids. To better understand cross sections, imagine that the solid is made up of moldable clay, and then imagine slicing it with a knife or string. The two-dimensional object seen on the sliced plane of the solid is known as a cross section. Cross sections can be quite complicated, depending on how the object is sliced. For simplicity, we will limit our discussion to the following: 1. All prisms and pyramids in this section have regular polygons as bases. 2. Cross sections will be limited in this section to those which are parallel or perpendicular to a base. 3. The solids in this section will be limited to the following: right prisms, right pyramids (pyramids in which the apex lies directly above the center of the base), right cylinders, and right cones. 4. In cross sections for prisms where the cross section is perpendicular to a base, the slicing will be performed through the center of the base, and also through the midpoint of one of the base edges. 5. In cross sections for pyramids where the cross section is perpendicular to the base, the slicing will be performed through the apex and also through the midpoint of one of the base edges. (This cross section will then also pass through the center of the base, since each pyramid in this section is a right pyramid with a regular polygon for a base.) Right Prisms Let us consider the following right prism and then draw some general conclusions. Initial Diagram Cross Section Parallel to the Bases Cross Section Perpendicular to the Bases We make the following observations which can be generalized to other right prisms: 1. In the cross section parallel to the bases, the cross section is congruent to the bases. 2. In the cross section perpendicular to the bases, the cross section is a rectangle.
14 Right Pyramids Let us consider the following right pyramid and then draw some general conclusions. Initial Diagram Cross Section Parallel to the Base Cross Section Perpendicular to the Base We make the following observations which can be generalized to other right pyramids: 1. In the cross section parallel to the base, the cross section is similar to the base. (In this particular diagram, the base is a square, and any cross section parallel to the base will result in a smaller square.) 2. In the cross section perpendicular to the base, the cross section is a triangle. Right Cylinders Let us consider the following right cylinder and then draw some general conclusions. Initial Diagram Cross Section Parallel to the Bases Cross Section Perpendicular to the Bases We make the following observations which can be generalized to other right cylinders: 1. In the cross section parallel to the bases, the cross section is a circle which is congruent to the bases. 2. In the cross section perpendicular to the bases, the cross section is a rectangle.
15 Right Cones Let us consider the following right cone and then draw some general conclusions. Initial Diagram Cross Section Parallel to the Base Cross Section Perpendicular to the Base We make the following observations which can be generalized to other right cones: 1. In the cross section parallel to the base, the cross section is another circle which is smaller than the base. 2. In the cross section perpendicular to the base, the cross section is a triangle. Exercises For each of the solids below, sketch two cross sections as described in the examples above. One cross section should be parallel to a base, and the other perpendicular to a base. Then identify each of the cross sections with a name (regular pentagon, triangle, rectangle, circle, etc.) 1. 2.
16 Exercises For each of the exercises below, sketch a solid which could have the given cross sections. (Some answers may not be unique.) 1. Cross section parallel to a base: Cross section perpendicular to a base:
17 2. Cross section parallel to a base: Cross section perpendicular to a base: 3. Cross section parallel to a base: Cross section perpendicular to a base: 4. Cross section parallel to a base: Cross section perpendicular to a base:
18 References Ballew, Pat. Images of the five Platonic Solids. Retrieved April 27, 2004, from
Characteristics of Solid Figures Face Edge Vertex (Vertices) A shape is characterized by its number of... Faces, Edges, and Vertices Faces - 6 Edges - 12 A CUBE has... Vertices - 8 Faces of a Prism The
CK-12 Geometry: Exploring Solids Learning Objectives Identify different types of solids and their parts. Use Euler s formula to solve problems. Draw and identify different views of solids. Draw and identify
Key Terms for This Session Session 9 Solids Previously Introduced congruent regular polygon vertex New in This Session cross section edge face net Platonic solid polyhedron Introduction In this session,
ENGINEERING DRAWING UNIT I - Part A 1. Solid Geometry is the study of graphic representation of solids of --------- dimensions on plane surfaces of ------- dimensions. 2. In the orthographic projection,
Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees Apex in a pyramid or cone, the vertex opposite the base; in
Level A 1. Which of the following is a 3-D shape? A) Cylinder B) Octagon C) Kite 2. What is another name for 3-D shapes? A) Polygon B) Polyhedron C) Point 3. A 3-D shape has four sides and a triangular
Geometry and Measurement of Solid Figures Activity Set 4 Trainer Guide Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF SOLID FIGURES
Geometry Vocabulary Booklet Geometry Vocabulary Word Everyday Expression Example Acute An angle less than 90 degrees. Adjacent Lying next to each other. Array Numbers, letter or shapes arranged in a rectangular
Constructing Geometric Solids www.everydaymathonline.com Objectives To provide practice identifying geometric solids given their properties; and to guide the construction of polyhedrons. epresentations
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
For more information - http://www.mathsisfun.com/geometry Begin recognition in EYFS Age related expectation at Y1 (secure use of language) shape, flat, curved, straight, round, hollow, solid, vertexvertices
3 D OBJECTS Properties of 3 D Objects A 3 Dimensional object (3 D) is a solid object that has 3 dimensions, i.e. length, width and height. They take up space. For example, a box has three dimensions, i.e.
10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres 10.4 Day 1 Warm-up 1. Which identifies the figure? A rectangular pyramid B rectangular prism C cube D square pyramid 3. A polyhedron
12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids
1 University of Waterloo Faculty of Mathematics Junior Math Circles March 10, 2010 3D Geometry II Centre for Education in Mathematics and Computing Opening Problem Three tennis ball are packed in a cylinder.
Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.
Nets of a cube puzzle Here are all eleven different nets for a cube, together with four that cannot be folded to make a cube. Can you find the four impostors? Pyramids Here is a square-based pyramid. A
MATH 206 - Midterm Exam 2 Practice Exam Solutions 1. Show two rays in the same plane that intersect at more than one point. Rays AB and BA intersect at all points from A to B. 2. If C is the midpoint of
UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,
Geometry Vocabulary Created by Dani Krejci referencing: http://mrsdell.org/geometry/vocabulary.html point An exact location in space, usually represented by a dot. A This is point A. line A straight path
. DISCOVERING 3D SHAPES WORKSHEETS OCTOBER-DECEMBER 2009 1 . Worksheet 1. Cut out and stick the shapes. SHAPES WHICH ROLL SHAPES WHICH SLIDE 2 . Worksheet 2: COMPLETE THE CHARTS Sphere, triangle, prism,
SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has
SHAPE NAMES Three-Dimensional Figures or Space Figures Rectangular Prism Cylinder Cone Sphere Two-Dimensional Figures or Plane Figures Square Rectangle Triangle Circle Name each shape. [triangle] [cone]
UNIT PLAN Grade Level: 2 Unit #: 5 Unit Name: Geometry Big Idea/Theme: Shapes can be analyzed, described, and related to our physical world. Culminating Assessment: Use real world objects from home, create
BOOK 8 ( CBSE / Chap10 /Visualizing Solid Shapes Chapter Facts. 1. A plane figure has two dimensions (2 D and a solid figure has three dimensions ( 3 D. 2. Some solid shapes consist of one figure embedded
Winter 2016 Math 213 Final Exam Name Instructions: Show ALL work. Simplify wherever possible. Clearly indicate your final answer. Problem Number Points Possible Score 1 25 2 25 3 25 4 25 Subtotal 100 Extra
Level: A 1. A 3-D shape can also be called... A) A flat shape B) A solid shape C) A polygon 2. Vertices are also called... A) Edges B) Corners C) Faces D) Sides 3. A solid object has six faces which are
ME 111: Engineering Drawing Lecture # 14 (10/10/2011) Development of Surfaces http://www.iitg.ernet.in/arindam.dey/me111.htm http://www.iitg.ernet.in/rkbc/me111.htm http://shilloi.iitg.ernet.in/~psr/ Indian
BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that
11-1 Space Figures and Cross Sections Vocabulary Review Complete each statement with the correct word from the list. edge edges vertex vertices 1. A(n) 9 is a segment that is formed by the intersections
6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-2008 Archimedean Solids Anna Anderson University
Name Class Date 17.1 Cross Sections and Solids of Rotation Essential Question: What tools can you use to visualize solid figures accurately? Explore G.10.A Identify the shapes of two-dimensional cross-sections
This publication is designed to support and enthuse primary trainees in Initial Teacher Training. It will provide them with the mathematical subject and pedagogic knowledge required to teach 3-D geometry
MENSURATION Definition 1. Mensuration : It is a branch of mathematics which deals with the lengths of lines, areas of surfaces and volumes of solids. 2. Plane Mensuration : It deals with the sides, perimeters
HOME LINK Line Segments, Rays, and Lines Family Note Help your child match each name below with the correct drawing of a line, ray, or line segment. Then observe as your child uses a straightedge to draw
MODULE THREE Shape and Space Fundamentals in ECD : Mathematics Literacy Learner s Manual MODULE THREE SHAPE AND SPACE Contents Unit One: Identifying Shapes 4 1. Angles 4 2. Triangles 5 3. Quadrilaterals
G3-33 Building Pyramids Goal: Students will build skeletons of pyramids and describe properties of pyramids. Prior Knowledge Required: Polygons: triangles, quadrilaterals, pentagons, hexagons Vocabulary:
Notes on polyhedra and 3-dimensional geometry Judith Roitman / Jeremy Martin April 23, 2013 1 Polyhedra Three-dimensional geometry is a very rich field; this is just a little taste of it. Our main protagonist
19. [Shapes] Skill 19.1 Comparing angles to a right angle. Place the corner of a page (which is a right angle) at the corner (vertex) of the angle. Align the base of the page with one line of the angle.
Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.
UNIT 7 Surface Area and Volume Managers of companies that produce food products must decide how to package their goods, which is not as simple as you might think. Many factors play into the decision of
MA.912.G.2 Geometry: Standard 2: Polygons - Students identify and describe polygons (triangles, quadrilaterals, pentagons, hexagons, etc.), using terms such as regular, convex, and concave. They find measures
V Viissuuaall & &P Peerrffoorrm miinngg A Arrttss P Prrooggrraam m,, S SJJU US SD D A Arrttss & &M Maatthh C Coonnnneeccttiioonnss Title/Description of Lesson Making a 3-dimensional sculpture using a 2dimensional
Geometry Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes
Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular
Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals
Chapter 18 Symmetry Symmetry is of interest in many areas, for example, art, design in general, and even the study of molecules. This chapter begins with a look at two types of symmetry of two-dimensional
Geometry Honors: Extending 2 Dimensions into 3 Dimensions Semester 2, Unit 5: Activity 30 Resources: SpringBoard- Geometry Online Resources: Geometry Springboard Text Unit Overview In this unit students
Surface Area of Rectangular & Right Prisms Surface Area of Pyramids Geometry Finding the surface area of a prism A prism is a rectangular solid with two congruent faces, called bases, that lie in parallel
Every 3-dimensional shape has three measurements to describe it: height, length and width. Height Width Length Faces A face is one of the flat sides of a three-dimensional shape. Faces A cuboid has 6 flat
1. Find the area of parallelogram ACD shown below if the measures of segments A, C, and DE are 6 units, 2 units, and 1 unit respectively and AED is a right angle. (a) 5 square units (b) 12 square units
II. Geometry and Measurement The Praxis II Middle School Content Examination emphasizes your ability to apply mathematical procedures and algorithms to solve a variety of problems that span multiple mathematics
Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base
24HourAnsers.com Online Homeork Focused Exercises for Math SAT Skill Set 12: Geometry - 2D Figures and 3D Solids Many of the problems in this exercise set came from The College Board, riters of the SAT
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
7.5 The student will a) describe volume and surface area of cylinders; b) solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c) describe how changing
Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 You can see why this works with the following diagrams: h h b b Solve: Find the area of
Students: 1. Students understand and compute volumes and areas of simple objects. *1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles. Review
Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose
Use of Exemplar 7 Exemplar Solids and Nets Objectives Dimension Learning Unit (1) To explore the 2-D representations of simple solids (2) To draw the net of a given solid Measures, Shape and Space Introduction
Florida Geometry EOC Assessment Study Guide The Florida Geometry End of Course Assessment is computer-based. During testing students will have access to the Algebra I/Geometry EOC Assessments Reference
Undergraduate Research Opportunity Programme in Science Polyhedra Name: Kavitha d/o Krishnan Supervisor: Associate Professor Helmer Aslaksen Department of Mathematics National University of Singapore 2001/2002
easures, hape & pace EXEMPLAR 7: 2-D Representation of Simple Solids Objective: To explore and sketch the 2-D representations of simple solids Key Stage: 3 Learning Unit: Introduction to Geometry Materials
Dear Grade 4 Families, During the next few weeks, our class will be exploring geometry. Through daily activities, we will explore the relationship between flat, two-dimensional figures and solid, three-dimensional
Integrated Algebra: Geometry Topics of Study: o Perimeter and Circumference o Area Shaded Area Composite Area o Volume o Surface Area o Relative Error Links to Useful Websites & Videos: o Perimeter and
Consolidation of Grade 6 EQAO Questions Geometry and Spatial Sense Compiled by Devika William-Yu (SE2 Math Coach) Overall Expectations GV1 Classify and construct polygons and angles GV2 GV3 Sketch three-dimensional
Target To know the properties of a rectangle (1) A rectangle is a 3-D shape. (2) A rectangle is the same as an oblong. (3) A rectangle is a quadrilateral. (4) Rectangles have four equal sides. (5) Rectangles
Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties
CHAPTER 12 Surface Area and Volume Chapter Outline 12.1 EXPLORING SOLIDS 12.2 SURFACE AREA OF PRISMS AND CYLINDERS 12.3 SURFACE AREA OF PYRAMIDS AND CONES 12.4 VOLUME OF PRISMS AND CYLINDERS 12.5 VOLUME
Name: Date: Period: Symmetry 1) Line Symmetry: A line of symmetry not only cuts a figure in, it creates a mirror image. In order to determine if a figure has line symmetry, a figure can be divided into
Name: Date: Geometry Honors 2013-2014 Solid Geometry Name: Teacher: Pd: Table of Contents DAY 1: SWBAT: Calculate the Volume of Prisms and Cylinders Pgs: 1-6 HW: Pgs: 7-10 DAY 2: SWBAT: Calculate the Volume
2.9 3-D Solids and Nets You need to decide whether or not to use the words shape (2-d) and solid (3-d) interchangeably. It can be helpful to avoid saying shape when referring to a 3-d object. Pupils will
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
Course: 7 th Grade Math DETAIL LESSON PLAN Wednesday, February 22 / Thursday, February 23 Student Objective (Obj. 3b) TSW identify three dimensional figures from nets. Lesson 8-9 Three-Dimensional Figures
MTH 139 FINL EXM REVIEW PROLEMS ring a protractor, compass and ruler. Note: This is NOT a practice exam. It is a collection of problems to help you review some of the material for the exam and to practice
2011-2012 Honors Geometry Final Exam Study Guide Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In each pair of triangles, parts are congruent as marked.
CHAPTER 6 - PRISMS AND ANTIPRISMS Prisms and antiprisms are simple yet attractive constructions, and form an interesting addition to the collection of geometric solids. They can adapt readily to unique
A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular
Tighes Hill Public School NSW Syllabus for the Australian Curriculum Measurement and Geometry Strand Three Dimensional Space Outcome Teaching and Learning Activities Notes/ Future Directions/Evaluation
1 University of Waterloo Faculty of Mathematics Grade 7/8 Math Circles Winter 2013 3D Geometry Introductory Problem Mary s mom bought a box of 60 cookies for Mary to bring to school. Mary decides to bring
Fill in the blank in each sentence with the vocabulary term that best completes the sentence. 1. A is a flat surface made up of points that extends infinitely in all directions. A plane is a flat surface