Raniasibaouihidn
Answered

Find detailed and accurate answers to your questions on IDNLearn.com. Discover trustworthy solutions to your questions quickly and accurately with help from our dedicated community of experts.

### 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.

Sagot :

GinnyAnsweridn
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.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions deserve precise answers. Thank you for visiting IDNLearn.com, and see you again soon for more helpful information.