Problem Analysis:
The task involves identifying the number of
vertices,
faces, and
edges for three different geometric figures. These are fundamental properties of polyhedra, and Euler's Formula can be used to verify the consistency of these counts. Euler's Formula states:
\[
V - E + F = 2
\]
where:
- \( V \) is the number of vertices,
- \( E \) is the number of edges,
- \( F \) is the number of faces.
We will solve each figure step by step.
---
Figure 1: Pyramid (Triangular Base)
#### Step 1: Identify the vertices (\( V \))
- The pyramid has a triangular base with 3 vertices.
- There is one additional vertex at the apex.
- Total vertices: \( V = 3 + 1 = 4 \).
#### Step 2: Identify the faces (\( F \))
- The base is a triangle, which is one face.
- There are 3 triangular lateral faces.
- Total faces: \( F = 1 + 3 = 4 \).
#### Step 3: Identify the edges (\( E \))
- The base is a triangle, contributing 3 edges.
- Each of the 3 lateral faces connects the apex to one of the base vertices, adding 3 more edges.
- Total edges: \( E = 3 + 3 = 6 \).
#### Verification Using Euler's Formula:
\[
V - E + F = 4 - 6 + 4 = 2
\]
The formula holds true.
#### Final Answer for Figure 1:
\[
\boxed{V = 4, F = 4, E = 6}
\]
---
Figure 2: Pentagonal Prism
#### Step 1: Identify the vertices (\( V \))
- The prism has two pentagonal bases.
- Each pentagon has 5 vertices.
- Total vertices: \( V = 5 + 5 = 10 \).
#### Step 2: Identify the faces (\( F \))
- There are 2 pentagonal faces (one on the top and one on the bottom).
- There are 5 rectangular lateral faces.
- Total faces: \( F = 2 + 5 = 7 \).
#### Step 3: Identify the edges (\( E \))
- Each pentagonal base has 5 edges.
- There are 5 vertical edges connecting corresponding vertices of the two bases.
- Total edges: \( E = 5 + 5 + 5 = 15 \).
#### Verification Using Euler's Formula:
\[
V - E + F = 10 - 15 + 7 = 2
\]
The formula holds true.
#### Final Answer for Figure 2:
\[
\boxed{V = 10, F = 7, E = 15}
\]
---
Figure 3: Cube
#### Step 1: Identify the vertices (\( V \))
- A cube has 8 vertices (one at each corner).
- Total vertices: \( V = 8 \).
#### Step 2: Identify the faces (\( F \))
- A cube has 6 square faces.
- Total faces: \( F = 6 \).
#### Step 3: Identify the edges (\( E \))
- Each face of the cube is a square, and there are 12 edges in total (each edge is shared by two faces).
- Total edges: \( E = 12 \).
#### Verification Using Euler's Formula:
\[
V - E + F = 8 - 12 + 6 = 2
\]
The formula holds true.
#### Final Answer for Figure 3:
\[
\boxed{V = 8, F = 6, E = 12}
\]
---
Final Summary of Answers:
1. Pyramid (Triangular Base): \( V = 4, F = 4, E = 6 \)
2. Pentagonal Prism: \( V = 10, F = 7, E = 15 \)
3. Cube: \( V = 8, F = 6, E = 12 \)
\[
\boxed{V = 4, F = 4, E = 6 \quad \text{(Pyramid)}, \quad V = 10, F = 7, E = 15 \quad \text{(Pentagonal Prism)}, \quad V = 8, F = 6, E = 12 \quad \text{(Cube)}}
\]
Parent Tip: Review the logic above to help your child master the concept of euler worksheet.
