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Question
Using Euler's formula find the unknown:
| Faces | ? | 5 | 20 |
| Vertices | 6 | ? | 12 |
| Edges | 12 | 9 | ? |
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Solution
\[\text { We know that the Euler's formula is: F+V = E+2 }\]
1 .\[\text { The number of vertices V is 6 and the number of edges E is 12 }.\]
\[\text { Using Euler's formula: }\]
\[\text { F+6 = 12+2 }\]
\[\text { F+6 = 14 }\]
\[\text { F = 14-6 }\]
\[\text { F = 8 }\]
\[\text { So, the number of faces in this polyhedron is 8 }.\]
2. \[\text { Faces, F = 5 }\]
\[\text { Edges, E = 9}.\]
\[\text { We have to find the number of vertices }.\]
\[\text { Putting these values in Euler's formula: }\]
\[\text { 5+V = 9+2 }\]
\[\text { 5+V = 11 }\]
\[\text { V = 11-5 }\]
\[\text { V = 6 }\]
\[\text { So, the number of vertices in this polyhedron is 6 }.\]
3. \[\text { Number of faces F = 20 }\]
\[\text { Number of vertices V = 12 }\]
\[\text { Using Euler's formula: }\]
\[\text { 20+12 = E+2 }\]
\[\text { 32 = E+2 }\]
\[\text { E+2 = 32 }\]
\[\text { E = 32-2 }\]
\[\text { E = 30 }.\]
\[\text { So, the number of edges in this polyhedron is 30 }.\]
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