PolyhedraPrisms and PyramidsOther Three Dimensional FiguresConclusionPolyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionMATH 113Section 8.3: Three-Dimensional FiguresProf. Jonathan DuncanWalla Walla UniversityWinter Quarter, 2008Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionOutline1Polyhedra2Prisms and Pyramids3Other Three Dimensional Figures4ConclusionPolyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionDefinitionsThe three-dimensional version of a polygon is called a polyhedron.PolyhedraA polyhedron is a simple closed surface composed entirely ofpolygon regions.ExampleAre all of the following polyhedra?Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionParts of a PolyhedraJust as with polygons, we can name the various parts of polyhedra.Parts of a PolyhedronThe surface of a polyhedron can be separated into various parts.FaceA face is one of the polygon regions on the surface of thepolyhedron.EdgeAn edge is a line segment connecting two adjacent polygon regions.VertexA vertex is a point connecting three or more edges.ExampleSketch a cube and identify its faces, edges, and vertices. What canyou say about the faces of the cube?Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionConnecting Polyhedra with PolygonsWe know several terms from our study of polygons. Can these beapplied to polyhedra as well?ExampleDecide if each term or concept below can be generalized topolyhedra and if it can, describe how.Simple Closed CurveSidesInterior AnglesClassification by the Number of SidesConvex and ConcaveRegularPolyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionRegular PolyhedraJust as we defined regular polygons in two dimensions, we candefine regular polyhedra in three.Regular PolyhedraA regular polyhedron is a convex polyhedron in which the faces arecongruent regular polygons and in which the number of edgeswhich meet at each vertex are the same.ExampleWhat is the simplest regular polyhedron you can think of?ExampleHow many regular polygons are there? How many regularpolyhedra are there?Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionThe Eight Platonic SolidsYou may find it surprising that while there are infinitely manyregular polygons, there are only 8 regular polyhedra.The Platonic SolidsThe eight regular polyhedra are often called the Platonic solids,named for the ancient Greek philosopher Plato who theorized thatthe classical elements were constructed from these regular solids.Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionKepler’s Model of the Solar SystemThese eight shapes held such meaning for early mathematicians andscientists that Johannes Kepler attempted to find a relationship betweenthe five known planets during his time and the five Platonic solids.Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionEuler’s DiscoveryBelow is a table with the number of vertices, edges, and faces inthe regular polyhedra.Vertices, Edges, and FacesVertices Edges FacesTetrahedron 4 6 4Cube 8 12 6Octahedron 6 12 8Dodecahedron 20 30 12Icosahedron12 30 20Euler’s FormulaThe relationship between these values is given by Euler’s formula:V + F = E + 2. This works for all polyhedra.Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionWhat is a PrismJust as we could divide polygons into families based on the numberof sides, we can divide polyhedra into families based on othercharacteristics.PrismsA polyhedron with two parallel bases what are congruent polygonsis called a prism. The non-base faces are called lateral faces.ExampleWhat one shape can be used the describe the lateral faces of allprisms?Prisms as Stacks of PolygonsAnother way to think of a prism is as many copies of the samepolygon stacked on top of one another.Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionTypes of PrismsThere are many different types of prisms. They can be classified inseveral ways.Clarification by BasePrisms can be classified by the type of polygon used as a base. Forexample, a triangular prism or a rectangular prism.Classification by Lateral FacesPrisms can be classified by their type of lateral faces. Prisms withrectangular lateral faces are called right prisms. Prisms which arenot right prisms are called oblique prisms.ExampleSketch a right pentagonal prism and an oblique rectangular prism.Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionPyramidsAnother interesting type of polyhedron is a pyramid. This term isbroader than what is usually thought of as a pyramid.PyramidsA pyramid is a polyhedron whose base is a polygon and whoseother faces are triangles with a common vertex. That commonvertex is called the apex of the pyramid.Pyramids and StackingAlternatively, a pyramid can be built by stacking smaller andsmaller similar polygons.ExampleHow do pyramids and prisms compare. In particular think about: basesand right vs. oblique pyramids and prisms.Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionFrom Prisms to CylindersIf we think of taking right prisms with n-gon bases and letting nincrease, the shape we approach is called a cylinder.CylindersA cylinder is a simple closed surface that is bounded by twocongruent circles that lie in parallel planes.ExampleWhat aspects or terms used to describe prisms can also be used todescribe cylinders?Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionFrom Pyramids to ConesIf we think of making pyramids with n-gon bases and letting nincrease, we approach a shape called a cone.ConesA cone is the collection of all line segments between a simpleclosed region of a plane (the base) and a point outside the plane(the apex).ExampleBy taking cross-sections of a right-circular cone, we can get severaldifferent two-dimensional shapes called conic sections. These are:CircleEllipseParabolaHyperbola (single side)Polyhedra Prisms and Pyramids Other Three Dimensional Figures ConclusionImportant ConceptsThings to Remember from Section 8.31Definition and parts of polyhedra2Euler’s Formula for Polyhedra3Definition and types of prisms4Definition and types of pyramids5Definitions and types of cones and