Mathematics
Faces Edges and Vertices
Faces, edges, and vertices are key elements of 3-dimensional shapes. Faces are the flat surfaces of a shape, edges are the lines where two faces meet, and vertices are the points where edges meet. Understanding these components helps in identifying and classifying different shapes, such as cubes, pyramids, and prisms.
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eBook - PDFDr. Math Introduces Geometry
Learning Geometry is Easy! Just ask Dr. Math!
- (Author)
- 2004(Publication Date)
- Jossey-Bass(Publisher)
108 Dr. Math Introduces Geometry These are made of two or more different types of regular polygons, all arranged in the same sequence around each vertex. So you might have a pentagon, a square, a triangle, and a square around each vertex, in that order. There aren’t any vertices where the order would be pentagon, triangle, triangle, square. There is also an interesting relationship among the number of faces (f ), edges (e), and vertices (v) of a polyhedron. The mathemati- cian Leonard Euler discovered that in every polyhedron, f – e + v = 2 For example, in a cube, f = 6, e = 12, and v = 8, and 6 – 12 + 8 = 2. —Dr. Math, The Math Forum Dear Leon, What an odd question. We generally use these terms in different settings. A cube on its own has six faces. Here we’re not picturing it set on a table but just sort of floating in space so that all six faces are equal, and we don’t think of any of them as special. When we are talking about how to calculate the area or the volume, we usually think of one face as the bottom and call it the base, as if we were setting it down on a table to measure it. The top may be seen as the other base, since the Introduction to Three-Dimensional (3-D) Geometric Figures 109 Dear Dr. Math, In math class we are learning about polyhe- dra, and I can’t figure out the difference between a base and a face on the shapes we are learning. What is the difference? How many bases does a cube have, and how many faces does it have? Yours truly, Leon Bases and Faces two sides are identical, and the other faces are the sides. So when you set the cube down, it has one base (or two if you prefer) and four sides. It really doesn’t make any difference which face you call the base when you talk about a cube, because the faces are all the same.
eBook - PDF- Tom Bassarear, Meg Moss(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
In the ramp, only the triangle faces are opposite. The other three faces are noncongruent rectangles. The numbers of faces, edges, and vertices are different. The cube: 6 faces, 12 edges, 8 vertices The ramp: 5 faces, 9 edges, 6 vertices INVESTIGATION 8.3e Discovering Euler’s Formula Let’s continue our look at faces, edges, and vertices. You can create three-dimensional models using raisins or gumdrops as the vertices and toothpicks for the edges. Build the following fig- ures and count the number of faces, edges, and vertices of each. Look for patterns and a formula showing the relationship between the faces, vertices, and edges of each. Cube, square-based pyramid, triangular-based prism, tetrahedron, octahedron DISCUSSION A fun and interesting thing to do with your models is to dip them into soapy water and see the designs made by the bubbles. There is a field of math that studies such things. First, compare what you found to these numbers to make sure you counted correctly, and then consider a formula. Polyhedron Faces Vertices Edges Cube 6 8 12 Square-based pyramid 5 5 8 Triangular-based prism 5 6 9 Tetrahedron 4 4 6 Octahedron 8 6 12 Section 8.3 Three-Dimensional Figures 503 n n Connecting Two-Dimensional Representations To Three-Dimensional Objects There are many ways in which the two-dimensional and three-dimensional worlds connect. All buildings, from small sheds to large skyscrapers, are designed before they are built. To enable the architects and the engineers to communicate, blueprints are designed and studied. So that the electricians, plumbers, and other members of the building team will know where to place the appropriate wires and fixtures, other kinds of drawings are used. Each of these drawings requires someone to think about the object in three dimensions and then represent that information two dimensionally, although computer simulation is changing the nature of these representations.
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
The sides of each of the faces are called edges ; for example, AB is an edge of the cube in Figure 8.108. The vertices of the polyhedron are simply the vertices of the polygonal regions that form the polyhedron; for example, E and F are vertices of the cube in Figure 8.108. Convex and concave Just as polygons can be convex or concave, so can polyhedra. Before reading the definition of a convex polyhedron, think back to the definition of a convex polygon and see whether you can modify that definition for three-dimensional objects. Then read on. . . . A polyhedron is convex if and only if any line segment connecting two points of the polyhedron is either on the surface or in the interior of the polyhedron (Figure 8.109). TABLE 8.5 Polygons Polyhedra (two-dimensional) (three-dimensional) What they are Simple closed curves Definition Union of line segments Component parts Vertices Line segments Angles Other? Classification By number of sides Regular vs. not regular Convex vs. concave Inside Surface Outside Figure 8.107 H G C B A E F D Figure 8.108 LANGUAGE Some mathematics dictionaries and textbooks define edge as the side of a polyhedron. Some focus on the term line segment. Yet others define edge as the intersection of two faces. On one mid-term exam, I asked my students to define edge. Interestingly, all three of these perspectives and interpretations appeared in the stu-dents’ definitions! Concave Convex Figure 8.109 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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