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

Hint: 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 \[{\text{ - 1}}\].