Question

Complete the table by using Euler’s formula.

F	8	5	?
v	6	?	12
E	?	9	30

Answer 1

To complete the table, we will make use of Euler’s formula. According to the Euler’s formula, we have the following relation:
$\text{F}+\text{V - E = }2$
Where F is the number of faces, V is the number of vertices and E is the number of edges.

Complete step-by-step answer:
Before solving the question, we must first know what Euler’s formula is. Euler’s formula is a topological invariance relating the number of faces, vertices and edges of any polyhedron. The relation between them is given by,
$\text{F}+\text{V - E = }2$
Where F is the number of faces, V is the number of vertices and E is the number of edges. In the table given, we are given the faces, vertices and edges of a polyhedron. In the first column, F denotes the number of faces, V denotes the number of vertices and E denotes the number of edges. In the second column, the number of faces given is 8 and the number of vertices given is 6 and we have to find the number of edges. So by Euler’s formula we have:

& \text{F + V - E = 2} \\\ & \Rightarrow 8\text{ }+\text{ }6\text{ - E = }2 \\\ & \Rightarrow 14\text{ - E = }2 \\\ & \Rightarrow 14\text{ }-\text{ }2\text{ = E} \\\ & \Rightarrow \text{E = }12 \\\ \end{aligned}$$ In the second column, the number of faces given is 5, the number of vertices are unknown and the number of edges are 9. By Euler’s formula, we have: $$\begin{aligned} & \text{F + V - E = 2} \\\ & \Rightarrow 5\text{ }+\text{ V - }9\text{ = }2 \\\ & \Rightarrow \text{V - 4 = }2 \\\ & \Rightarrow \text{V = }6 \\\ \end{aligned}$$ In the third column, the number of vertices given is 12, the number of edges are 30 and the number of faces are to be determined. So by Euler’s formula, we have: $$\begin{aligned} & \text{F + V - E = }2 \\\ & \Rightarrow \text{F + }12\text{ }-\text{ }30\text{ = }2 \\\ & \Rightarrow \text{F - }18\text{ = }2 \\\ & \Rightarrow \text{F = }20 \\\ \end{aligned}$$ Thus, the completed table is shown below: F| 8| 5| 20 ---|---|---|--- v| 6| 6| 12 E| 12| 9| 30 **Note:** The Euler’s formula, which we have used to calculate the value of F, V and E is not applicable in all the cases. This formula is invalid in cases when the polyhedrons intersect themselves. Thus, while solving the question, we have assumed that the polyhedrons are non-intersecting.