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Worked examples include cube-like computations and specific solids such as an octahedron, hexagonal prism, icosahedron, and polyhedra made from triangular, pentagonal, and hexagonal 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is Euler's formula for polyhedrons and what do V, F, and E mean?","Question",{"text":63,"@type":64},"Euler's formula is V + F − E = 2, where V is the number of vertices, F is the number of faces, and E is the number of edges.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How can you relate the total number of face sides to the number of edges?",{"text":68,"@type":64},"The worksheet uses total sides = 2 × number of edges because each face side corresponds to an edge shared by two faces.",{"name":70,"@type":61,"acceptedAnswer":71},"What minimum number of faces is needed to close a polyhedron?",{"text":72,"@type":64},"The minimum number of faces and vertices is 4 to close a polyhedron, analogous to needing three line segments to close a polygon.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},274686,1789481442,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & 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objects with Euler's characteristic 2 can be morphed into which of the following shapes?\u003Cbr>2) What is the relationship between the number of sides of all the faces in a polyhedron to the number of edges?\u003Cbr>3) What is the minimum number of faces in a polyhedron?\u003Cbr>4) Is it possible to have a polyhedron with 7 edges?\u003Cbr>5) An octahedron has 8 faces and 12 edges. How many vertices does it have?\u003Cbr>6) What is the value ofF + V − E for a hexagonal prism?\u003Cbr>7) An icosahedron has 30 edges and 12 vertices. How many faces does it have?\u003Cbr>8) A polyhedron is made up of 12 triangular faces. How many vertices does the polyhedron have?\u003Cbr>9) A polyhedron is made up of 6 pentagonal faces. How many vertices does the polyhedron have?\u003Cbr>10) A polyhedron is made up of 10 haxagonal faces. How many vertices does the polyhedron have? |\n\nName: Date:  \n\n| ANSWERS :\u003Cbr>\u003Cbr>Euler's Formula |\n| --- |\n| Answer the following questions.\u003Cbr>1) Three dimensional objects with Euler's characteristic 2 can be morphed into which of the following shapes? Spheres\u003Cbr>2) What is the relationship between the number of sides of all the faces in a polyhedron to the number of edges? Total \\# Sides = \\# Polygons × n = 2 × \\# Edges. Every polygon face shares one edge with another polygon in the polyhedron.\u003Cbr>3) What is the minimum number of faces in a polyhedron? 4\u003Cbr>4) Is it possible to have a polyhedron with 7 edges? No\u003Cbr>5) An octahedron has 8 faces and 12 edges. How many vertices does it have?\u003Cbr>V + F-E = 2\u003Cbr>V + 8-12 = 2 V = 2 + 12-8 = 6\u003Cbr>6) What is the value ofF + V − E for a hexagonal prism? 2 Euler's Formula is true for any polyhedron.\u003Cbr>7) An icosahedron has 30 edges and 12 vertices. How many faces does it have?\u003Cbr>12 + F-30 = 2 F = 2 + 30-12 = 20 faces\u003Cbr>8) A polyhedron is made up of 12 triangular faces. How many vertices does the polyhedron have?\u003Cbr>12 triangles × 3 sides = 36 sides which equals 36/2 = 18 edges. V + 12-18 = 2 V = 20-12 = 8 vertices\u003Cbr>9) A polyhedron is made up of 6 pentagonal faces. How many vertices does the polyhedron have? 6 pentagons × 5 sides = 30 sides which equals 30/2 = 15 edges. V + 6-15 = 2 V = 17-6 = 11 vertices\u003Cbr>10) A polyhedron is made up of 10 hexagonal faces. How many vertices does the polyhedron have? 10 hexagons × 6 sides = 60 sides which equals 60/2 = 30 edges. V + 10-30 = 2 V = 32-10 = 22 vertices |\n\nKEY CONCEPTS:  \nEuler's Formula (for Polyhedrons)  \nV + F-E = 2 V= \\# Vertices, F = \\# Faces, E = \\# Edges  \nApplies for simple or non-intersecting 3-D objects including polyhedrons referred to as objects with Euler characteristic 2.  \ne.g. Cube = 8 + 6-12 = 2  \nPolyhedron: Closed three dimensional object with polygonal faces.  \n• The minimum number of faces and vertices is 4 to close a polyhedron (much like 3 lines are needed to close a polygon)  \n•  The number of edges, E, is half the number of sides (E= \\# polygon faces × n/2)  \n• Objects with Euler characteristic 2 can be morphed into spheres. Objects with different Euler characteristics change into different shapes. Objects with characteristic 0, for example, morph into a torus or doughnut shape.","cbCainb4sGFTBJEa","https://ap.wps.com/l/cbCainb4sGFTBJEa","pdf",160192,"English","# Euler's Formula for Polyhedrons\n## Core Relationship: V + F − E = 2\n## Questions and Answer Calculations\n## Key Concepts and Applications","[{\"question\":\"What is Euler's formula for polyhedrons and what do V, F, and E mean?\",\"answer\":\"Euler's formula is V + F − E = 2, where V is the number of vertices, F is the number of faces, and E is the number of edges.\"},{\"question\":\"How can you relate the total number of face sides to the number of edges?\",\"answer\":\"The worksheet uses total sides = 2 × number of edges because each face side corresponds to an edge shared by two faces.\"},{\"question\":\"What minimum number of faces is needed to close a polyhedron?\",\"answer\":\"The minimum number of faces and vertices is 4 to close a polyhedron, analogous to needing three line segments to close a polygon.\"}]","Euler's Formula Worksheet - Polyhedrons - Answer Practice | PDF"]