{"id":132,"date":"2017-08-23T09:03:13","date_gmt":"2017-08-23T09:03:13","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=132"},"modified":"2017-09-18T16:42:58","modified_gmt":"2017-09-18T16:42:58","slug":"classifying-two-and-three-dimensional-solids","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/classifying-two-and-three-dimensional-solids\/","title":{"rendered":"Classifying Two- and Three-Dimensional Solids"},"content":{"rendered":"
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\u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

<\/p>\n

Classifying Two- and Three-Dimensional Solids<\/h1>\n

Objective<\/h4>\n

In this lesson, you will study the classification of two- and three-dimensional solids using geometric properties\u00a0such as number of vertices, edges, and faces.<\/p>\n

Previously Covered:<\/h4>\n

How do we classify two-dimensional objects?<\/strong><\/h3>\n

The fundamental two-dimensional objects are polygons.\u00a0Informally, a polygon is a figure created by fitting together a\u00a0certain number of line segments.<\/p>\n

A polygon<\/abbr> is the union of n <\/em>line\u00a0segments P1,\u00a0<\/sub>P2<\/sub><\/em>,\u00a0P2,\u00a0<\/sub>P 3<\/sub><\/em>,\u00a0. . . P3,\u00a0<\/sub>P1<\/sub><\/em> such that P1<\/sub>,\u00a0P2<\/sub>, . . . Pn\u00a0<\/sub><\/em>is a sequence of n\u00a0<\/em>distinct points\u00a0in the plane, with <\/em>,\u00a0satisfying the following properties:<\/p>\n

(1) No two segments intersect except at their\u00a0endpoints.<\/em><\/p>\n

(2) No two segments with a common endpoint are\u00a0collinear.
\n<\/em><\/p>\n

A polygon with n\u00a0<\/em>sides is called an n<\/em>-gon<\/strong>.<\/p>\n

\n

Question<\/h4>\n
\n

Which of these figures is not a polygon?<\/p>\n

    \n
  1. \"Trapezoid\"<\/li>\n
  2. \"Two<\/li>\n
  3. \"Plus<\/li>\n
  4. \"Arrowhead\"<\/li>\n<\/ol>\n<\/div>\n

    Reveal Answer <\/a><\/p>\n

    \n

    The correct answer is B. This figure does not satisfy\u00a0condition (1) in the definition of polygon.<\/p>\n<\/div>\n<\/section>\n

    We can distinguish between polygons by counting the number of\u00a0sides, or equivalently, the number of corners. This allows us to\u00a0distinguish between different \u201cshapes.\u201d Below are\u00a0examples of a 3-gon (or triangle) and a 4-gon (or quadrilateral).<\/p>\n

    \"Triangle\"<\/p>\n

    \"Quadrilateral\"<\/p>\n

    A polygon is convex if no two points of it lie on opposite sides of a line containing\u00a0a side of the polygon or more simplistically a polygon such that a\u00a0line containing the side of the polygon does not cross the\u00a0interior of the polygon. Polygons which are not convex are called\u00a0concave polygons.<\/strong><\/p>\n

    \n

    Question<\/h4>\n
    \n

    Which of the figures below is not a convex pentagon?<\/p>\n

      \n
    1. \"Concave<\/li>\n
    2. \"regular<\/li>\n
    3. \"Pentagon<\/li>\n
    4. \"Pentagon<\/li>\n<\/ol>\n<\/div>\n

      Reveal Answer <\/a><\/p>\n

      \n

      The correct answer is A. If we extend one of the two bottom\u00a0sides, we can find points lying on opposite sides of an extended\u00a0line or the extensions cross the interior of the polygon.<\/p>\n<\/div>\n<\/section>\n

      In the example above, all the convex figures have the more or\u00a0less the same shape. So when we think about classifying figures\u00a0by shape, we are really thinking about convex figures.<\/p>\n

      \n

      The most perfect shapes are called regular figures<\/em>.<\/h3>\n

      A polygon is regular<\/abbr> if it is convex, all sides are congruent<\/abbr>,\u00a0and all angles are congruent. A regular octagon is shown below.<\/p>\n

      <\/p>\n

      \n

      Important Tidbit<\/h4>\n

      We can construct regular n<\/em>-sided polygons for any positive integer n<\/em>, where n<\/em> is greater than or equal to 3. So there are infinitely many distinct regular polygons.<\/p>\n<\/div>\n

      How do we classify three-dimensional objects?<\/strong><\/h3>\n

      The analogue of a polygon in three dimensions is a polyhedron<\/em>.\u00a0A polyhedron<\/abbr> is a three-dimensional figure made of up sides that are polygons.\u00a0A solid\u00a0polyhedron<\/abbr> is the union of a polyhedron and its\u00a0interior. The polygons are referred to as faces<\/abbr>.\u00a0The faces intersect in line segments called edges<\/abbr> and the edges meet at points called vertices<\/abbr>.<\/p>\n

      A convex\u00a0polyhedron<\/abbr> is a polyhedron such that no two points of\u00a0it lie on opposite sides of a plane containing a face of the\u00a0polyhedron. This implies that all faces of the polyhedron are\u00a0convex polygons, Hence the polyhedron to the right below is\u00a0convex. Another way to think of a convex polyhedron is that when\u00a0you connect any two points on the polyhedron the segment formed\u00a0must be contained on the polyhedron or in its interior.<\/p>\n

      \"Figures<\/center>\"Figures<\/p>\n

      The simplest types of polyhedra are prisms. A prism<\/abbr> is a polyhedron with two congruent polygonal faces contained in\u00a0parallel planes. These faces are called the bases<\/abbr> of the prism.<\/p>\n

      We can distinguish between two prisms by analyzing their bases.<\/p>\n

      \n

      Question<\/h4>\n
      \n

      Which of the following is not a hexagonal prism?<\/p>\n

        \n
      1. \"hexagonal<\/li>\n
      2. \"hexagonal<\/li>\n
      3. \"hexagonal<\/li>\n
      4. \"hexagonal<\/li>\n<\/ol>\n<\/div>\n

        Reveal Answer <\/a><\/p>\n

        \n

        The correct answer is D . This is a pentagonal prism.<\/p>\n<\/div>\n<\/section>\n

        How do we distinguish between three-dimensional solids that are not prisms?<\/strong><\/h3>\n

        As was true for prisms, we can distinguish between different types of polyhedra\u00a0by analyzing the faces and counting the faces, vertices, and edges.<\/p>\n

        Suppose we are given two three-dimensional solids that are not prisms. Can we\u00a0distinguish between a tetrahedron and a pyramid just by examining the shape of the faces? Sometimes\u00a0we can.<\/p>\n

        \"figures<\/p>\n

        A pyramid has one face that is a square, while the tetrahedron has triangles as\u00a0faces.<\/p>\n

        We will need a bit more information to compare the hexahedron with an\u00a0octahedron because both figures have only triangular faces. The polygon to the left below is the\u00a0octahedron.<\/p>\n\n\n\n
        \n
        \"tetrahedron\"<\/div>\n<\/td>\n
        		\n
        \"octahedron\"<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        A count of the number of faces, edges, and vertices along with the shape of a\u00a0face is usually enough to distinguish between any two solids.<\/p>\n

        \n

        Question<\/h4>\n
        \n

        For example, use this table to record the number of faces,\u00a0edges, and vertices of a tetrahedron, cube, and triangular prism.<\/p>\n\n\n\n\n\n\n\n\n
        \n
        Vertices, Edges, and Faces of Solids<\/div>\n<\/th>\n<\/tr>\n
        <\/th>\n\n
        Vertices<\/div>\n<\/th>\n
        		\n
        Edges<\/div>\n<\/th>\n
        		\n
        Faces<\/div>\n<\/th>\n<\/tr>\n<\/thead>\n
        Tetrahedron<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
        Cube<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
        Triangular prism<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

        Reveal Answer <\/a><\/p>\n

        \n\n\n\n\n\n\n\n\n
        \n
        Vertices, Edges, and Faces of Solids<\/div>\n<\/th>\n<\/tr>\n
        <\/th>\n\n
        Vertices<\/div>\n<\/th>\n
        		\n
        Edges<\/div>\n<\/th>\n
        		\n
        Faces<\/div>\n<\/th>\n<\/tr>\n<\/thead>\n
        Tetrahedron<\/td>\n4<\/strong><\/td>\n6<\/strong><\/td>\n4<\/strong><\/td>\n<\/tr>\n
        Cube<\/td>\n8<\/strong><\/td>\n12<\/strong><\/td>\n6<\/strong><\/td>\n<\/tr>\n
        Triangular prism<\/td>\n6<\/strong><\/td>\n9<\/strong><\/td>\n5<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/section>\n

        Notice that there is a relationship between the number of faces, edges, and\u00a0vertices. In fact, any polyhedron satisfies Euler\u2019s Formula: , where F<\/em> = number of faces, E<\/em> = number of edges, and\u00a0V<\/em> = number of vertices.<\/p>\n

        \n

        Important Tidbit<\/h4>\n

        A polyhedron is regular<\/strong> if all its faces are congruent regular polygons,\u00a0all of its dihedral angles are congruent, and the same number of faces meet at each vertex.\u00a0The problem of classifying regular polyhedra is a very old problem, one studied extensively\u00a0by the Greek philosopher Plato. The answer to this question is surprising in contrast to the\u00a0analog in two dimensions. There are only five regular polyhedra, and they are referred to as\u00a0the Platonic solids.<\/p>\n\n\n\n<\/colgroup>\n\n\n\n\n
        \n

        \"<\/p>\n<\/td>\n

        		\n

        \"<\/p>\n<\/td>\n<\/tr>\n

        \n

        \"<\/p>\n<\/td>\n

        		\n

        \"<\/p>\n<\/td>\n<\/tr>\n

        \n

        \"<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/section>\n

        <\/p>\n

        \u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

        Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

        \u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Classifying Two- and Three-Dimensional Solids Objective In this lesson, you will study the classification of two- and three-dimensional solids using geometric properties\u00a0such as number of vertices, edges, and faces. Previously Covered: How do we classify two-dimensional objects? The fundamental two-dimensional objects are polygons.\u00a0Informally, a polygon is a figure created […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-132","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/132","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=132"}],"version-history":[{"count":14,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/132\/revisions"}],"predecessor-version":[{"id":750,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/132\/revisions\/750"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}