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### 9.1: Faces, Vertices, and Edges

In an earlier lesson, you saw the equation [tex]\( V + F - 2 = E \)[/tex], which relates the number of vertices, faces, and edges in a Platonic solid.

1. Write an equation that makes it easier to find the number of vertices in each of the Platonic solids described:
a. An octahedron, which has 8 faces.
b. An icosahedron, which has 30 edges.

2. A Buckminsterfullerene (also called a "Buckyball") is a polyhedron with 60 vertices. It is not a Platonic solid, but the numbers of faces, edges, and vertices are related the same way as those in a Platonic solid. Write an equation that makes it easier to find the number of faces a Buckyball has if we know how many edges it has.



Answer :

GinnyAnswer
Certainly! Let's address each part of the question step by step.

### 1. Finding the Number of Vertices using [tex]\( V + F - 2 = E \)[/tex]

#### a. Octahedron (8 faces):

Given:
- Faces ([tex]\( F \)[/tex]) = 8

We start with the equation [tex]\( V + F - 2 = E \)[/tex].

To find the number of vertices ([tex]\( V \)[/tex]):

1. Rearrange the equation to find [tex]\( E \)[/tex]:
[tex]\[ V + F - 2 = E \Rightarrow E = 2F - 4 \][/tex]

2. Substituting [tex]\( F \)[/tex] with 8:
[tex]\[ E = 2(8) - 4 = 16 - 4 = 12 \][/tex]

3. Now, substitute [tex]\( E \)[/tex] back into the original equation to solve for [tex]\( V \)[/tex]:
[tex]\[ V + 8 - 2 = 12 \][/tex]
[tex]\[ V + 6 = 12 \][/tex]
[tex]\[ V = 12 - 6 \][/tex]
[tex]\[ V = 6 \][/tex]

Therefore, an octahedron has 6 vertices.

#### b. Icosahedron (30 edges):

Given:
- Edges ([tex]\( E \)[/tex]) = 30

We start with the equation [tex]\( V + F - 2 = E \)[/tex].

To find the number of vertices ([tex]\( V \)[/tex]):

1. Let’s find a relationship for Faces ([tex]\( F \)[/tex]) first. From the equation:
[tex]\[ 2(V + F - 2) = 2E \Rightarrow 2V + 2F - 4 = 2E \Rightarrow 3F = 2E \][/tex]
[tex]\[ F = \frac{2E}{3} \][/tex]

2. Substituting [tex]\( E \)[/tex] with 30:
[tex]\[ F = \frac{2(30)}{3} = \frac{60}{3} = 20 \][/tex]

3. Now, substitute [tex]\( F \)[/tex] and [tex]\( E \)[/tex] back into the original equation to solve for [tex]\( V \)[/tex]:
[tex]\[ V + 20 - 2 = 30 \][/tex]
[tex]\[ V + 18 = 30 \][/tex]
[tex]\[ V = 30 - 18 \][/tex]
[tex]\[ V = 12 \][/tex]

Therefore, an icosahedron has 12 vertices.

### 2. Finding the Number of Faces for a Buckyball:

Given:
- Vertices ([tex]\( V \)[/tex]) = 60

We start with the equation [tex]\( V + F - 2 = E \)[/tex].

To find the number of faces ([tex]\( F \)[/tex]) if we know the number of edges ([tex]\( E \)[/tex]):

1. Rearrange the equation to express [tex]\( F \)[/tex]:
[tex]\[ F = E - V + 2 \][/tex]

Therefore, the equation to find the number of faces ([tex]\( F \)[/tex]) for a Buckyball given the number of edges ([tex]\( E \)[/tex]) is:
[tex]\[ F = E - 60 + 2 \][/tex]

Using this formula, if, for example, the Buckyball has 90 edges, we find the number of faces as follows:
[tex]\[ F = 90 - 60 + 2 = 32 \][/tex]

Therefore, a Buckyball with 90 edges has 32 faces.

The results are:
- An octahedron has 6 vertices.
- An icosahedron has 12 vertices.
- A Buckyball with 90 edges has 32 faces.

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