Question

Using Euler’s formula find the unknown.

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

Answer 1

Hint: First of all, write the number of faces, edges and vertices in every given part. Then, substitute the given values in the Euler’s formula, which states that the summation of the number of faces and vertices is exactly two more than the number of edges. Then, solve the equations to find the value of unknowns.

Complete step by step solution: Euler’s formula gives us the relationship between the number of faces, the number of edges and the number of faces of a solid figure.
We know that Euler’s formula states that [Number of faces+ Number of vertices- Number of edges =2]
Which can be written as $F + V - E = 2$, where $F$ represents the number of faces, $V$ represents the number of vertices and $E$ represents the number of edges.
Here, in the first part, we are given that the number of vertices is 5 and the number of edges is 12. We have to determine the number of faces using Euler’s formula.
We will substitute the values of the number of vertices and edges in the Euler’s formula.
Hence, we will have $F + 6 - 12 = 2$
On solving the above equation, we will get,
$F - 6 = 2$
Adding 6 to both sides,
$F = 8$
Thus, the number of faces is 8.
In the second part, we have the number of faces as 5 and the number of edges as 9. We have to find the number of vertices using Euler’s formula.
Here, $F = 5$ and $E = 9$
Thus, on substituting the values in Euler’s formula, we will get,
$5 + V - 9 = 2$
On solving the expression, we will get,
$V - 4 = 2$
Adding 4 on both sides,
$V = 6$
Thus, the number of vertices is 6.
In the third part, we have the number of faces as 20 and the number of edges as 12. We have to calculate the number of edges using Euler’s formula.
Here, $F = 20$ and $V = 12$
On substituting these values in Euler’s formula , $F + V - E = 2$, we will get,
$20 + 12 - E = 2$
On solving the above expression, we will get,
$
  32 - E = 2 \\
   \Rightarrow 32 - 2 = E \\
   \Rightarrow E = 30 \\
$
Hence, the number of edges is 30.
Therefore, we can fill the values of unknowns as

Faces	8	5	20
Vertices	6	6	12
Edges	12	9	30

Note: Euler’s formula holds true for any convex polyhedrons. Students may make mistakes in interpreting values of the number of faces, edges and vertices from the given table. Euler’s formula does not work for the shapes with holes or when it intersects itself.