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.
6 Key excerpts on "Faces Edges and Vertices"
eBook - ePub- Christine Hopkins, Ann Pope, Sandy Pepperell(Authors)
- 2013(Publication Date)
- David Fulton Publishers(Publisher)
...It is also not uncommon for children to use the names for plane shapes when describing 3-D objects (e.g. square instead of cube, and rectangle instead of cuboid). 3-D objects have faces, vertices and edges. We do not use the word side when describing 3-D shapes as it can be confusing. (We often use sides in 2-D when we mean the outer edge of a polygon; in 3-D the outside is made up of surfaces.) In mathematics a face is any surface of the 3-D object, an edge is where two faces meet, and a vertex (plural vertices) is a point where edges meet. A solid with plane (or flat) faces only A solid with a curved face and two plane faces Think about 3-D shapes you know. Can you think of a shape that has no edges or vertices? A shape that has no vertices? How many edges does it have? Can you think of a shape that has just one edge? A sphere has just one curved face, with no edges or vertices. If you cut through the centre of a sphere you will get a hemisphere – one flat face, one curved face and just one edge. A cylinder has one curved face, two plane faces which are circles and two edges. A cone has one vertex, one edge, a curved face and a flat face. In general an edge is formed wherever two faces meet. POLYHEDRA Polyhedra are 3-D shapes that have flat or plane faces only; this means that the faces are polygons. Cubes and cuboids are everyday examples, as are prisms and pyramids. Cardboard shapes and materials like Polydron or Clixi are excellent for exploring polyhedra. The naming for polyhedra is similar to that for polygons. ‘Poly’ means many and ‘hedron’ means surface (or literally seat). Look at the vertex of a polyhedron...
eBook - ePub- Tony Cotton(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...The most common 3D shapes are shown as follows. One way of classifying 3D shapes is by the numbers of faces, edges and vertices (singular vertex). The faces of a shape are the flat regions. The edges are where two faces meet, and a vertex is a point at which two or more edges meet. Position When we want to define, or locate, a point mathematically, we use coordinates on a pair of axes (you pronounce this ‘axees’, not like the implement you chop wood with). These axes cross each other at what is known as the origin, which has coordinates (0,0). The four areas created by the axes are known as the four quadrants. The best way to explain these terms is by a diagram. In the following diagram the x-axis is the horizontal axis and the y-axis is the vertical axis: The two axes are labelled x and y. We write coordinates as a pair of numbers in brackets separated by a comma, for example (2,3) or (21,4). The first number always refers to the x-coordinate and the second number refers to the y-coordinate. Primary pupils are also introduced to defining direction by compass points. There are four points of the compass: north, south, east and west. The direction halfway between north and west is described as northwest. Similarly, the direction halfway between south and east is described as southeast, the direction halfway between south and west is southwest, and the direction halfway between north and east is northeast. Once we have placed a shape on a grid, we can change its shape through a transformation : this is the process of moving a shape on a grid. The three transformations are translation, reflection and rotation. A translation leaves the shape’s dimensions and its orientation unchanged. In other words, it is the same as sliding the same shape across the grid...
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
...4 Geometry Focal Points Undefined Terms Angles Simple Closed Curves, Regions, and Polygons Circles Constructions Third Dimension Coordinate Geometry Transformations and Symmetry You might be surprised about how many real-life concepts are included in the study of geometry. Young children experiment with ideas such as over versus under, first versus last, right versus left, and between, without realizing that they are studying important mathematics concepts. Additionally, early attempts at logic, even the common everyone else gets to do it arguments that are so popular with children, are geometry topics. A cube is a three-dimensional object with six congruent faces and eight vertices—often called a block. You may also use the term block to identify a prism that is not a cube, but, although you may not think of it very often, you can probably identify the differences between a cube and a prism that is not a cube, as shown in Fig. 4.1. Fig. 4.1. In this chapter, you will review, refine, and perhaps, extend your understanding of geometry. When Euclid completed a series of 13 books called the Elements in 300 BC, he provided a logical development of geometry that is unequaled in our history and is the foundation of our modern geometry study. Geometry is a dynamic, growing, and changing body of intuitive knowledge. We will let you explore conjectures and provide opportunities for you to create informal definitions. There will be some reliance on terms and previous knowledge, especially when we get to standard formulas. Undefined Terms Some fundamental concepts in geometry defy definition. If we try to define point, space, line, and plane, then we find ourselves engaging in circular (flawed) logic. The best we can do is accept these fundamental concepts as building blocks and try to explain them. A fixed location is called a point, which is a geometric abstraction that has no dimension, only position. We often use a tiny round dot as a representation of a point...
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Solid lines indicate edges that you actually see, and dashed lines indicate edges that are obscured from the two-dimensional perspective. Edges are the lines where two faces (flat surfaces) meet on a three-dimensional figure. On the figure shown below, the “main” face is shaded and the arrows indicate which way the figure is facing. It takes a while to get used to this way of looking at three-dimensional figures. The GED ® test formula sheet provides the formulas you will need to do surface area and volume problems with three-dimensional figures. Still, you need to know what these terms mean as well as what the variables in the given equation mean or the formulas won’t be of any use to you. Therefore, this section on three-dimensional figures provides information to help you understand the GED ® test formula sheet, but it isn’t necessary to memorize the formulas. The surface area of a three-dimensional figure is exactly what it sounds like. It is the total area of all the faces, even those you cannot see in the picture. So for surface area, we need to remember the formulas for the areas of the two-dimensional faces that make up each three-dimensional figure. Again, these are given on the GED ® test formula sheet. The surface area of a rectangular solid can be thought of as the area of wrapping paper that covers a shirt box with no overlapping. Volume is how much the three-dimensional figure can hold. It is sometimes called capacity. Basically, for three-dimensional figures that have identical “tops” and “bottoms” (bases), it is the area of the base (bottom or top) multiplied by the height of the figure. Note the dimensions for each measure. Although feet are shown here, you can substitute “inches,” “centimeters,” or whatever the problem is using. Note that the power (exponent) matches the number of the dimensions of the figure. Rectangular Prisms One type of three-dimensional figure is known as a rectangular prism, which is also referred to as a right prism...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 6 Geometry Topics CHAPTER 6 GEOMETRY TOPICS Plane geometry refers to two-dimensional shapes (that is, shapes that can be drawn on a sheet of paper), such as triangles, parallelograms, trapezoids, and circles. Three-dimensional objects (that is, shapes with depth) are the subjects of solid geometry. TRIANGLES A closed three-sided geometric figure is called a triangle. The points of the intersection of the sides of a triangle are called the vertices of the triangle. A side of a triangle is a line segment whose endpoints are the vertices of two angles of the triangle. The perimeter of a triangle is the sum of the measures of the sides of the triangle. An interior angle of a triangle is an angle formed by two sides and includes the third side within its collection of points. The sum of the measures of the interior angles of a triangle is 180°. A scalene triangle has no equal sides. An isosceles triangle has at least two equal sides. The third side is called the base of the triangle, and the base angles (the angles opposite the equal sides) are equal. An equilateral triangle has all three sides equal.. An equilateral triangle is also equiangular, with each angle equaling 60°. An acute triangle has three acute angles (less than 90°). An obtuse triangle has one obtuse angle (greater than 90°). A right triangle has a right angle. The side opposite the right angle in a right triangle is called the hypotenuse of the right triangle. The other two sides are called the legs (or arms) of the right triangle. By the Pythagorean Theorem, the lengths of the three sides of a right triangle are related by the formula c 2 = a 2 + b 2 where c is the hypotenuse and a and b are the other two sides (the legs). The Pythagorean Theorem is discussed in more detail in the next section. An altitude, or height, of a triangle is a line segment from a vertex of the triangle perpendicular to the opposite side...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...All lateral faces are triangles. EXAMPLES: Pyramids and prisms are named by the type of base they have. The base is the “first name,” and then pyramid or prism is the “last name.” A prism with a base of a rectangle is called a rectangular prism. A pyramid with a base of a pentagon is called a pentagonal pyramid. Some shapes do not have vertices, edges, or faces. These shapes have specific names. DEFINITIONS Cylinder A solid with two circular bases that are congruent and parallel. Cone A solid with one circular base; the points along the circle are joined at a point outside the circle. Sphere A solid where every point on the surface is the same distance from the center; a three-dimensional object shaped like a ball. Nets of Three-Dimensional Shapes The net is what a 3D shape looks like if it is opened out flat or unfolded. Think about constructing a paper box. The net can then be folded up to make the shape. Here are some examples. EXAMPLE 11.6 Name each of the solids. 1) 2) 3) 4) 5) 6) 7) 8) SOLUTIONS 1) cube 2) rectangular prism 3) triangular prism 4) triangular pyramid 5) square pyramid 6) cone 7) cylinder 8) sphere 11.7 What Are Surface Area and Volume and How Do We Calculate Them? DEFINITION Surface area The sum of the area of each of the surfaces. It is helpful to think of the net of the shape when trying to picture surface area. For example, if you were to paint all the faces of a rectangular prism, the surface area would be the paint. Rectangular prism What is the surface area of a rectangular prism with a height of 10 in, length of 12 in, and width of 7 in? Write the formula: SA = 2 lw + 2 lh + 2 wh Substitute: SA = 2(12)(7) + 2(12)(10) + 2(7)(10) Calculate: SA = 168 + 240 +...
