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Polyhedra VEF Polyhedra KEY Polyhedra (Polyhedron, singular) are 3-D objects with faces that are polygons. Doing math is just like paradise. ☺ (F) Faces are the flat smooth surfaces. The faces are 2-D polygons. (E) Edges are where two faces meet. (V) Vertices are the “points” where edges meet. NOTE: Remember to keep the vocabulary straight for our formulas! For example, a “six-sided die” is considered to be a cube with 6 faces , NOT 6 sides! We will now introduce a new vocabulary phrase “ sides of faces ”. This is different from a “Face” or from an “Edge”. When viewing a polygon face, think of it as a 2-D object. Therefore, we can view the sides of the polygon as “sides of faces”. “Side of a face” is different than an “Edge” since an “Edge” is a 3-D label. In fact, each “Edge” is composed of two “sides of faces” because each edge is where two faces meet. If we go around a polyhedron and count the total number of “sides of faces” from all of its polygon faces, then we have in fact counted all of the edges of the polyhedron twice! This gives us the following formula: Total Sides of Faces = 2E . Together with Euler’s famous formula for polyhedra, V – E + F = 2 , we are able to determine the number of edges and vertices of any polyhedron just by knowing the number and type of polygon faces that it has. NOTE: we consider only convex polyhedra. 1 Polyhedra KEY Total Sides of Faces = 2E V – E + F = 2 1. A cube has six faces which are all squares. How many edges and vertices does it have? type of face Square # of each face 6 # of sides per face 4 total sides of faces 24 Total Sides of Faces 2E 24 2E 12 E V – E F 2 V – 12 6 2 V – 6 2 V 8 A cube has 12 edges and 8 vertices. 2. Dice are often used in gaming such as the role-playing game Dungeons & Dragons . The 20- sided die is a type of polyhedron called an icosahedron. It has 20 faces all of which are equilateral triangles. How many edges and vertices does an icosahedron have? type of face Triangle # of each face 20 # of sides per face 3 total sides of faces 60 Total Sides of Faces 2E 60 2E 30 E V – E F 2 V – 30 20 2 V – 10 2 V 12 An icosahedron has 30 edges and 12 vertices. 2 Polyhedra KEY 3. Suppose a convex polyhedron has 11 faces, of which 4 are triangles, 5 are quadrilaterals, and 2 are hexagons. How many vertices and edges does it have? type of face Triangle Quadrilateral Hexagon # of each face 4 5 2 # of sides per face 3 4 6 total sides of faces 12 20 12 Total Sides of Faces 2E 122012 2E 44 2E 22 E V – E F 2 V – 22 11 2 V – 11 2 V 13 The convex polyhedron has 22 edges and 13 vertices. 3 Polyhedra KEY 4. The cuboctahedron is a polyhedron that can be constructed by slicing off the corners of a cube through the midpoints of the edges as shown below. a. How many triangle faces does a cuboctahedron have? The cuboctahedron has one triangle face for each vertex from the previous cube. Since the cube has 8 vertices (see question #1) then it follows that the cuboctahedron has 8 triangle faces. b. How many square faces does a cuboctahedron have? The cuboctahedron has one square face for each left-over face from the previous cube. Since the cube has 6 faces (see question #1) then it follows that the cuboctahedron has 6 square faces. c. How many edges and vertices does a cuboctahedron have? type of face Triangle Square # of each face 8 6 # of sides per face 3 4 total sides of faces 24 24 Total Sides of Faces 2E 24 24 2E 48 2E 24 E NOTE: F 8 6 14 V – E F 2 V – 24 14 2 V – 10 2 V 12 The cuboctahedron has 24 edges and 12 vertices. 4 Polyhedra KEY ADDITIONAL EXERCISES 5. A “net” for a polyhedron is a squashed version that shows, in a 2-D way, how the polygon faces are connected to each other. In fact, if you were to cut out the net image from this piece of paper, you could fold it back up into the 3-D polyhedron! To the right is the net for the Pentagonal Rotunda. You can see that it has 17 faces, of which 6 are pentagons, 10 are triangles, and 1 is a decagon. How many edges and vertices does it have? type of face Pentagon Triangle Decagon # of each face 6 10 1 # of sides per face 5 3 10 total sides of faces 30 30 10 Total Sides of Faces 2E 30 30 10 2E 70 2E 35 E V – E F 2 V – 35 17 2 V – 18 2 V 20 The Pentagonal Rotunda has 35 edges and 20 vertices. 5 Polyhedra KEY 6. Below is the net and image for the Augmented Dodecahedron. How many edges and vertices does it have? type of face Pentagon Triangle # of each face 11 5 # of sides per face 5 3 total sides of faces 55 15 Total Sides of Faces 2E 55 15 2E 70 2E 35 E NOTE: F 11 5 16 V – E F 2 V – 35 16 2 V – 19 2 V 21 The Augmented Dodecahedron has 35 edges and 21 vertices. 6 Polyhedra KEY 7. Below is the net for the Pentagonal Orthocupolarontunda. How many edges and vertices does it have? type of face Triangle Square Pentagon # of each face 15 5 7 # of sides per face 3 4 5 total sides of faces 45 20 35 Total Sides of Faces 2E 45 20 35 2E 100 2E 50 E NOTE: F 15 5 7 27 V – E F 2 V – 50 27 2 V – 23 2 V 25 The Pentagonal Orthocupolarontunda has 50 edges and 25 vertices. 7 Polyhedra KEY 8. Below is the net and image for the Hexagonal Antiprism. How many edges and vertices does it have? type of face Triangle Hexagon # of each face 12 2 # of sides per face 3 6 total sides of faces 36 12 Total Sides of Faces 2E 36 12 2E 48 2E 24 E NOTE: F 12 2 14 V – E F 2 V – 24 14 2 V – 10 2 V 12 The Hexagonal Antiprism has 24 edges and 12 vertices. 8 Polyhedra KEY 9. The 8-sided die is a type of polyhedron called an octahedron. It is a polyhedron with eight triangle faces. How many vertices and edges does it have? type of face Triangle # of each face 8 # of sides per face 3 total sides of faces 24 Total Sides of Faces 2E 24 2E 12 E V – E F 2 V – 12 8 2 V – 4 2 V 6 An octahedron has 12 edges and 6 vertices. 9 Polyhedra KEY 10. The truncated octahedron can be constructed by slicing off the corners of an octahedron so that a square is formed. a. How many square faces does a truncated octahedron have? The truncated octahedron has one square face for each vertex from the original octahedron. Since an octahedron has 6 vertices (see question #9) then it follows that the truncated octahedron has 6 square faces. b. How many hexagon faces does a truncated octahedron have? The truncated octahedron has one hexagon face for each left-over face from the original octahedron. Since an octahedron has 8 faces (see question #9) then it follows that the truncated octahedron has 8 hexagon faces. c. How many edges and vertices does a truncated octahedron have? type of face Square Hexagon # of each face 6 8 # of sides per face 4 6 total sides of faces 24 48 Total Sides of Faces 2E 24 48 2E 72 2E 36 E NOTE: F 6 8 14 V – E F 2 V – 36 14 2 V – 22 2 V 24 The truncated octahedron has 36 edges and 24 vertices. 10 Polyhedra KEY 11. The truncated icosahedron can be constructed by slicing off the corners of an icosahedron so that a pentagon is formed. a. How many pentagon faces does the truncated icosahedron have? The truncated icosahedron has one pentagon face for each vertex from the original icosahedron. Since an icosahedron has 12 vertices (see question #2) then it follows that the truncated icosahedron has 12 pentagon faces. b. How many hexagon faces does the truncated icosahedron have? The truncated icosahedron has one hexagon face for each left-over face from the original icosahedron. Since an icosahedron has 20 faces (see question #2) then it follows that the truncated icosahedron has 20 hexagon faces. c. How many edges and vertices does the truncated icosahedron have? type of face Pentagon Hexagon # of each face 12 20 # of sides per face 5 6 total sides of faces 60 120 Total Sides of Faces 2E 60 120 2E 180 2E 90 E NOTE: F 12 20 32 V – E F 2 V – 90 32 2 V – 58 2 V 60 The truncated icosahedron has 90 edges and 60 vertices. 11 Polyhedra KEY 12. The Trapezoidal Hexecontahedron is a polyhedron with 60 quadrilateral faces. How many vertices and edges does it have? type of face Quadrilateral # of each face 60 # of sides per face 4 total sides of faces 240 Total Sides of Faces 2E 240 2E 120 E V – E F 2 V – 120 60 2 V – 60 2 V 62 The Trapezoidal Hexecontahedron has 120 edges and 62 vertices.
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