Question

Using Euler's formula find the unknown:

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

Answer 1

$$\begin{gathered} \text{We know that the Euler's formula is: F+V = E+2} \\ \text{(i)} \\ \text{The number of vertices V is 6 and the number of edges E is 12.} \\ \text{Using Euler's formula:} \\ \text{F+6 = 12+2} \\ \text{F+6 = 14} \\ \text{F = 14-6} \\ \text{F = 8} \\ \text{So, the number of faces in this polyhedron is 8.} \\ \text{(ii)} \\ \text{Faces, F = 5} \\ \mathrm{E}\text{dges, E = 9.} \\ \mathrm{W}\text{e have to find the number of vertices.} \\ \text{Putting these values in Euler's formula:} \\ \text{5+V = 9+2} \\ \text{5+V = 11} \\ \text{V = 11-5} \\ \text{V = 6} \\ \text{So, the number of vertices in this polyhedron is 6.} \\ \text{(iii)} \\ \mathrm{N}\text{umber of faces F = 20} \\ \mathrm{N}\text{umber of vertices V = 12} \\ \text{Using Euler's formula:} \\ \text{20+12 = E+2} \\ \text{32 = E+2} \\ \text{E+2 = 32} \\ \text{E = 32-2} \\ \text{E = 30.} \\ \text{So, the number of edges in this polyhedron is 30.} \end{gathered}$$