Math Problem Statement

For the planar graph shown, calculate the number of interior faces, total faces, and apply Euler's formula for planar graphs.

Solution

For the planar graph shown:

  1. Number of interior faces: Count the regions inside the graph that are enclosed by edges. The graph has 3 interior faces.

  2. Total number of faces: In a planar graph, the total number of faces includes all interior faces plus the exterior face. Therefore, the graph has 4 faces in total.

  3. Euler's formula: Euler's formula for planar graphs is VE+F=2V - E + F = 2, where VV is the number of vertices, EE is the number of edges, and FF is the number of faces.

    For this graph:

    • V=6V = 6 (vertices: A, B, C, D, E, F)
    • E=8E = 8 (edges)

    So, substituting into Euler's formula, we get: 68+4=26 - 8 + 4 = 2

  4. Faces for another planar graph: Without the graph shown, it’s not possible to calculate how many faces another graph would have without further details.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Planar Graphs
Euler's Formula
Graph Theory

Formulas

Euler's Formula: V - E + F = 2

Theorems

Euler's Formula for Planar Graphs

Suitable Grade Level

Grades 10-12