IDENTIFY THE NUMBER OF EDGES VERTICES AND FACES USING EULERS RULE

Problem 1 :

Using Euler's rule, determine the number of edges for a figure with 5 vertices and 4 regions.

Solution :

Number of vertices = 5, number of regions = 4

E = V + R - 2

E = 5 + 4 - 2

E = 9 - 2

E = 7

Problem 2 :

Using Euler's rule, determine the number of edges for a figure with 6 vertices and 5 regions.

Solution :

Number of vertices = 6, number of regions = 5

E = V + R - 2

E = 6 + 5 - 2

E = 11 - 2

E = 9

Problem 3 :

Using Euler's rule, determine the number of vertices for a figure with 7 edges and 3 regions

Solution :

Number of edges = 7, number of regions = 3 

E = V + R - 2

7 = V + 3 - 2

7 = V + 1

V = 7 - 1

V = 6

Problem 4 :

Using Euler's rule, determine the number of vertices for a figure with 9 edges and 4 regions

Solution :

Number of edges = 9, number of regions = 4 

E = V + R - 2

9 = V + 4 - 2

9 = V + 2

V = 9 - 2

V = 7

Problem 5 :

Using Euler's rule, determine the number of regions for a figure with 10 edges and 8 vertices.

Solution :

Number of edges = 10, number of vertices = 8

E = V + R - 2

10 = 8 + R - 2

10 = 6 + R

R = 10 - 6

R = 4

Problem 6 :

Using Euler's rule, determine the number of regions for a figure with 12 edges and 7 vertices.

Solution :

Number of edges = 12, number of vertices = 7

E = V + R - 2

12 = 7 + R - 2

12 = 5 + R

R = 12 - 5

R = 7

Problem 7 :

Use Euler's rule

E = V + R - 2

to determine the number of :

a) Edges in a plane figure with 11 vertices and 5 faces.

b) Faces in a plane figure with 9 vertices and 17 edges.

Solution :

E = V + F - 2

a) Number of vertices = 11, number of faces = 5

E = 11 + 5 - 2

E = 16 - 2

E = 14

b)  Number of vertices = 9, number of edges = 17

E = V + F - 2

17 = 9 + F - 2

17 = 7 + F

F = 17 - 7

F = 10

Problem 8 :

Use Euler's rule E = V + F - 2 to determine the number of :

a) Vertices if there are 7 faces and 14 edges.

b)  faces if there are 12 edges and 8 vertices.

Solution :

a) F = 7, E = 14

E = V + F - 2

14 = V + 7 - 2

14 = V + 5

V = 14 - 5

V = 9

b) V = 8, E = 12

E = V + F - 2

12 = 8 + F - 2

12 = F + 6

F = 12 - 6

F = 6