https://file.aiquickdraw.com/mathbot/file/1730237275606o3a9nftg.jpg

To find the number of regions (or faces) that a planar graph divides the plane into, we can use **Euler's formula** for planar graphs:

\[
V - E + F = 2
\]

where:
- \( V \) is the number of vertices,
- \( E \) is the number of edges, and
- \( F \) is the number of faces.

Let's analyze the graph given in the image:

1. **Count the vertices** (\( V \)): There are 8 vertices labeled from 1 to 8.
   
2. **Count the edges** (\( E \)): We can count each line connecting two vertices. This graph appears to be a **complete graph** with 8 vertices (\( K_8 \)), where each pair of vertices is connected by a unique edge. In a complete graph, the number of edges can be found using the formula:
   \[
   E = \frac{V(V-1)}{2} = \frac{8 \cdot 7}{2} = 28
   \]

3. **Calculate the faces** (\( F \)): Substitute \( V = 8 \) and \( E = 28 \) into Euler's formula:
   \[
   8 - 28 + F = 2
   \]
   \[
   F = 2 + 28 - 8 = 22
   \]

Thus, the graph divides the plane into **22 faces** when drawn with no intersecting edges.

Would you like more details or have any questions?

### Related Questions
1. How is Euler's formula derived for planar graphs?
2. What are planar graphs, and why are they significant in graph theory?
3. How do we verify if a graph is planar?
4. How would Euler's formula apply to a non-planar graph?
5. Can the formula be extended to three-dimensional graphs or polyhedra?

### Tip
In complete graphs, if \( V > 4 \), the graph might not be planar because it will require intersecting edges unless modified. Euler's formula only applies without intersections.

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

Learn how to use Euler's formula to determine the number of faces created by a planar graph with 8 vertices and 28 edges. This guide covers key concepts in graph theory, such as planar graphs, and includes a detailed step-by-step solution. Suitable for high school students in Grades 10-12.

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