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.