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Using Euler's formula, how many edges does a polyhedron with 6 faces and 8 vertices have?

Euler's Formula: [tex] F + V = E + 2 [/tex]

[tex] F = 6 [/tex]
[tex] V = 8 [/tex]
[tex] E = ? [/tex]

How many edges does the polyhedron have?

[tex] \text{Edges:} \, E = ? [/tex]

Sagot :

GinnyAnsweridn
To determine the number of edges a polyhedron has using Euler's formula, we follow these steps:

1. Identify the known values:
- Number of faces, [tex]\( F = 6 \)[/tex]
- Number of vertices, [tex]\( V = 8 \)[/tex]

2. State Euler’s formula:
[tex]\[ F + V = E + 2 \][/tex]
This formula relates the number of faces ([tex]\( F \)[/tex]), vertices ([tex]\( V \)[/tex]), and edges ([tex]\( E \)[/tex]) of a polyhedron.

3. Substitute the known values into Euler’s formula:
[tex]\[ 6 + 8 = E + 2 \][/tex]

4. Solve for [tex]\( E \)[/tex] (the number of edges):
[tex]\[ 14 = E + 2 \][/tex]

5. Isolate [tex]\( E \)[/tex] by subtracting 2 from both sides of the equation:
[tex]\[ 14 - 2 = E \][/tex]

6. Simplify the equation:
[tex]\[ E = 12 \][/tex]

Therefore, a polyhedron with 6 faces and 8 vertices has [tex]\( 12 \)[/tex] edges.
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