https://file.aiquickdraw.com/mathbot/file/1730239654689jh3fbtod.jpg

To find the number of faces (regions) in a planar graph, 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 (regions).

### Step 1: Count the Vertices and Edges
In this graph:
- **Vertices (V)**: There are 8 vertices, numbered 1 through 8.
- **Edges (E)**: Counting the lines between vertices in the graph, there are 18 edges.

### Step 2: Use Euler's Formula
Plugging in these values:
\[
8 - 18 + F = 2
\]

Simplifying:
\[
F = 2 + 18 - 8
\]
\[
F = 12
\]

### Answer
The number of faces (regions) the graph cuts up the plane into is **12**.

Would you like further details or have any other questions?

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**Related Questions to Explore:**
1. What happens to Euler’s formula if a graph is not planar?
2. How would the number of faces change if more edges were added to this graph?
3. Can Euler’s formula be used for graphs in three dimensions, and if so, how?
4. What are the conditions that make a graph planar?
5. How does a change in the number of vertices affect the possible configurations of faces?

**Tip:** Euler’s formula is a powerful tool for analyzing planar graphs, and it’s especially useful in verifying whether a drawing is indeed planar if applied correctly.

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 find the number of regions (faces) a planar graph creates on a plane using Euler's formula. This problem involves counting vertices and edges, then applying V - E + F = 2 to determine the number of faces. Ideal for college-level students studying Graph Theory.

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