Question

If ${\text{F}} = 4$ and ${\text{V}} = 3$, then the value of ${\text{E}}$ is : (Use Euler’s formula)
A. $7$
B. $5$
C. $4$
D. $1$

Answer 1

Here we will be using Euler’s Formula which states that ${\text{F + V = E + 2}}$ where, ${\text{F}}$ is the number of faces, ${\text{V}}$ the number of vertices, and ${\text{E}}$ the number of edges.

Complete step-by-step solution:
Step 1: By substituting the values of ${\text{F}} = 4$ and ${\text{V}} = 3$ in the Euler’s formula
${\text{F + V = E + 2}}$ we get:${\text{4 + 3 = E + 2}}$
Step 2: By adding the terms in the RHS side of the expression ${\text{4 + 3 = E + 2}}$ we get:
$\Rightarrow 7{\text{ = E + 2}}$
By bringing ${\text{E}}$ into the RHS side and
${\text{7}}$ into the LHS side of the above expression we get:
$\Rightarrow {\text{E = 7 - 2}}$
By subtracting the terms in the RHS side of the above expression we get:
$\Rightarrow {\text{E = 5}}$
The value of ${\text{E}}$ is $5$.

Option B is the correct answer.

Note: Students should remember Euler’s formula which is topological invariance related to the number of faces, vertices, and edges of any polyhedron. It is written as ${\text{F + V = E + 2}}$ where ${\text{F}}$ is the number of faces, ${\text{V}}$ the number of vertices, and ${\text{E}}$ the number of edges.
For example, a cube has six faces, eight vertices, and twelve edges so it satisfies this formula because ${\text{6 + 8 = 12 + 2}}$ so the LHS side equals the RHS side.
The second Euler’s formula used in trigonometry says that ${{\text{e}}^{ix}} = \cos x + i\sin x$ where ${\text{e}}$ is the base of the natural logarithm and $i$ is the square root of ${\text{ - 1}}$.