Question

Verify Euler’s formula for the following three-dimensional figures:

Answer 1

(i) 

Number of vertices = 6
Number of faces = 8
Number of edges = 12
Using Euler formula,
F + V – E = 2
8 + 6 – 12 = 2
2 = 2 Hence proved.

(ii) 

Number of vertices = 9
Number of faces = 8
Number of edges = 15
Using, Euler’s formula,
F + V – E = 2
9 + 8 – 15 = 2
2 = 2 Hence proved.

(iii) 

Number of vertices = 9
Number of faces = 5
Number of edges = 12
Using, Euler’s formula,
F + V – E = 2
9 + 5 – 12 = 2
2 = 2 Hence proved.

Answer 2

Euler's formula for three-dimensional polyhedra is given by: V − E + F = 2

• V = Number of vertices,
• E = Number of edges,
• F = Number of faces.

Shape 1: Double Pyramid (Octahedron)

1. Vertices (V): 6 (4 vertices on the middle square + 2 vertices at the top and bottom tips).
2. Edges (E): 12 (4 edges on the middle square + 4 edges connecting the top vertex to the square + 4 edges connecting the bottom vertex to the square).
3. Faces (F): 8 (4 triangular faces on the top + 4 triangular faces on the bottom).

V − E + F

= 6 − 12 + 8 = 2

Shape 2: Square Pyramid

1. Vertices (V): 5 (4 vertices on the square base + 1 vertex at the top tip).
2. Edges (E): 8 (4 edges of the square base + 4 edges connecting the top vertex to each vertex of the square).
3. Faces (F): 5 (4 triangular faces + 1 square base).

V − E + F

= 5 − 8 + 5 = 2

Shape 3: Triangular Prism

1. Vertices (V): 6 (3 vertices on each triangular base).
2. Edges (E): 9 (3 edges on each triangular base + 3 edges connecting the corresponding vertices of the two triangles).
3. Faces (F): 5 (2 triangular bases + 3 rectangular faces connecting the sides of the triangles).

V − E + F

= 6 − 9 + 5 = 2

Verified.