Question

A octahedron has _____ faces and 6 vertices with 12 edges. (Use Euler’s formula)
\[\begin{align}
  & (\text{A) 4} \\
 & \text{(B) 8} \\
 & \text{(C) 12} \\
 & \text{(D) 24} \\
\end{align}\]

Answer 1

Hint: In the question, we were given that the number of edges of octahedron and number of vertices of an octahedron. Let us assume the number of faces in octahedron as F, number of vertices as V, number of faces as F, and number of edges as E. Now by Euler’s rule, \[F+V=E+2\] we will find the value of F. This value of F gives the number of faces of the octahedron.

Complete step-by-step solution:
Before solving the question, we should have a clear idea about Euler’s rule.
According to Euler’s Rule, “If F is the number of faces of a polyhedron, V be the number of polyhedron and E be the number of edges in a polyhedron, then \[F+V=E+2\]”.

From the question, we were given the number of vertices on the octahedron and the number of edges of the octahedron. With these, we have to find the number of faces of the octahedron.
The number of faces of octahedron can be found by Euler’s rule.
Let us assume the number of faces in octahedron as F, the number of vertices as V, and the number of edges as E.
So,
\[F+V=E+2\]
We have to find the number of faces.
So,
We should rewrite the formula such that we should place the number of faces on L.H.S and remaining terms on R.H.S.
\[F=E+2-V....(1)\]
In the question, we were given that number of edges of the octahedron is equal to 12.
So,
\[E=12......(2)\]
In the question, we were also given that the number of vertices on an octahedron is equal to 6.
So,
\[V=6.....(3)\]
Now we will substitute equation (2) and equation (3) in equation (1).
\[\begin{align}
  & \Rightarrow F=12+2-6 \\
 & \Rightarrow F=8.....(4) \\
\end{align}\]
So, the number of faces in an octahedron is equal to 8.
Hence, option B is correct.

Note: We should know that Euler’s rule deals with polyhedrons. By using Euler’s rule, we can say whether a polyhedron exists in nature or not. We should know that the value of F, V, and E should not be negative or zero at any cost. So, the value of F, V, and E should be positive. By using Euler’s rule, we can find whether a polyhedron of given values of F, V, and E exists or not.