Question

Using Euler's formula find the unknown:

Faces	?	5	20
Vertices	6	?	12
Edges	12	9	?

Answer 1

\[\text { We know that the Euler's formula is: F+V = E+2 }\]

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. \[\text { Number of faces F = 20 }\]

\[\text { Number of vertices V = 12 }\]

\[\text { Using Euler's formula: }\]

\[\text { 20+12 = E+2 }\]

\[\text { 32 = E+2 }\]

\[\text { E+2 = 32 }\]

\[\text { E = 32-2 }\]

\[\text { E = 30 }.\]

\[\text { So, the number of edges in this polyhedron is 30 }.\]