https://file.aiquickdraw.com/mathbot/file/1730235376613wbw9iavf.jpg

To solve this problem, we can use **Euler's Formula** for planar graphs, which states:

\[
V - E + F = 2
\]

where:
- \( V \) is the number of vertices,
- \( E \) is the number of edges,
- \( F \) is the number of faces (regions), which includes the outer region.

### Step-by-Step Solution

1. **Count the Vertices (V)**: The graph has 8 vertices, numbered from 1 to 8, so \( V = 8 \).

2. **Count the Edges (E)**: This is a complete graph with 8 vertices, often denoted as \( K_8 \). For a complete graph with \( n \) vertices, the number of edges \( E \) is given by:
   \[
   E = \frac{n(n-1)}{2}
   \]
   Substituting \( n = 8 \):
   \[
   E = \frac{8 \times 7}{2} = 28
   \]

3. **Apply Euler's Formula**:
   \[
   F = 2 - V + E = 2 - 8 + 28
   \]
   \[
   F = 22
   \]

### Answer
The graph divides the plane into **22 faces**.

Would you like further details or explanations?

### Related Questions
1. How would you apply Euler's formula for a non-planar graph?
2. What happens to Euler's formula when there are overlapping or intersecting edges?
3. How many edges are there in a complete graph with 10 vertices?
4. What is the significance of Euler's formula in topology?
5. How do we determine if a graph is planar?

### Tip
Remember, Euler's formula \( V - E + F = 2 \) only holds for connected planar graphs without intersecting edges.

The following graph is planar. (Trust me.) If it is drawn with no intersecting edges, into how many regions ("faces") does it cut up the plane?

Learn how to calculate the number of faces in a planar graph using Euler's formula. This problem involves a complete graph with 8 vertices and requires counting edges and vertices to determine the number of regions. Suitable for high school students studying graph theory.

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