Answer :
Certainly! To identify the names of solid objects and to calculate their faces, edges, vertices, and verify the Euler's formula [tex]\( F + V - E = 2 \)[/tex], you need to understand the individual properties of each solid. Let's look at some common examples:
a) Cube (or Regular Hexahedron)
- Faces (F): A cube has 6 faces. Each face is a square.
- Edges (E): A cube has 12 edges.
- Vertices (V): A cube has 8 vertices.
- Verification of Euler's formula: [tex]\( F + V - E = 6 + 8 - 12 = 2 \)[/tex].
b) Tetrahedron
- Faces (F): A tetrahedron has 4 faces. Each face is a triangle.
- Edges (E): A tetrahedron has 6 edges.
- Vertices (V): A tetrahedron has 4 vertices.
- Verification of Euler's formula: [tex]\( F + V - E = 4 + 4 - 6 = 2 \)[/tex].
c) Octahedron
- Faces (F): An octahedron has 8 faces. Each face is an equilateral triangle.
- Edges (E): An octahedron has 12 edges.
- Vertices (V): An octahedron has 6 vertices.
- Verification of Euler's formula: [tex]\( F + V - E = 8 + 6 - 12 = 2 \)[/tex].
d) Dodecahedron
- Faces (F): A dodecahedron has 12 faces. Each face is a regular pentagon.
- Edges (E): A dodecahedron has 30 edges.
- Vertices (V): A dodecahedron has 20 vertices.
- Verification of Euler's formula: [tex]\( F + V - E = 12 + 20 - 30 = 2 \)[/tex].
Each polyhedron description includes:
1. Name: The specific solid we're examining.
2. Faces, Edges, and Vertices: Defined by counting the respective elements.
3. Verification: Ensuring that Euler's formula [tex]\( F + V - E = 2 \)[/tex] holds true for the solid.
One can use these principles to establish and confirm the properties for other solid objects. If you encounter a new or different polyhedron, remember to count the faces, edges, and vertices, and apply Euler's formula to verify the consistency of these properties.
a) Cube (or Regular Hexahedron)
- Faces (F): A cube has 6 faces. Each face is a square.
- Edges (E): A cube has 12 edges.
- Vertices (V): A cube has 8 vertices.
- Verification of Euler's formula: [tex]\( F + V - E = 6 + 8 - 12 = 2 \)[/tex].
b) Tetrahedron
- Faces (F): A tetrahedron has 4 faces. Each face is a triangle.
- Edges (E): A tetrahedron has 6 edges.
- Vertices (V): A tetrahedron has 4 vertices.
- Verification of Euler's formula: [tex]\( F + V - E = 4 + 4 - 6 = 2 \)[/tex].
c) Octahedron
- Faces (F): An octahedron has 8 faces. Each face is an equilateral triangle.
- Edges (E): An octahedron has 12 edges.
- Vertices (V): An octahedron has 6 vertices.
- Verification of Euler's formula: [tex]\( F + V - E = 8 + 6 - 12 = 2 \)[/tex].
d) Dodecahedron
- Faces (F): A dodecahedron has 12 faces. Each face is a regular pentagon.
- Edges (E): A dodecahedron has 30 edges.
- Vertices (V): A dodecahedron has 20 vertices.
- Verification of Euler's formula: [tex]\( F + V - E = 12 + 20 - 30 = 2 \)[/tex].
Each polyhedron description includes:
1. Name: The specific solid we're examining.
2. Faces, Edges, and Vertices: Defined by counting the respective elements.
3. Verification: Ensuring that Euler's formula [tex]\( F + V - E = 2 \)[/tex] holds true for the solid.
One can use these principles to establish and confirm the properties for other solid objects. If you encounter a new or different polyhedron, remember to count the faces, edges, and vertices, and apply Euler's formula to verify the consistency of these properties.
