What is the number of vertices in a dodecahedron?

What is the number of vertices in a dodecahedron?

• A. 12
• B. 20
• C. 24
• D. 30

Answer: (B)

A dodecahedron is one of the five Platonic solids and is characterized by having 12 faces, each of which is a regular pentagon. To determine the number of vertices, we can use Euler's formula, which states that for any convex polyhedron, the relationship between the number of vertices (V), edges (E), and faces (F) is given by: \[ V - E + F = 2 \] In the case of a dodecahedron: - The number of faces (F) is 12. - Each face is a pentagon, and there are 5 edges per face. However, since each edge is shared between two faces, the total number of edges (E) can be calculated as \( E = \frac{5 \times 12}{2} = 30\). Now, substituting the known values into Euler's formula gives us: \[ V - 30 + 12 = 2 \] Simplifying this equation: \[ V - 18 = 2 \] Adding 18 to both sides results in: \[ V = 20 \] Thus, a dodecahedron has 20 vertices. This is why the correct answer is recognized as the number

A dodecahedron is one of the five Platonic solids and is characterized by having 12 faces, each of which is a regular pentagon. To determine the number of vertices, we can use Euler's formula, which states that for any convex polyhedron, the relationship between the number of vertices (V), edges (E), and faces (F) is given by:

[ V - E + F = 2 ]

In the case of a dodecahedron:

• The number of faces (F) is 12.

• Each face is a pentagon, and there are 5 edges per face. However, since each edge is shared between two faces, the total number of edges (E) can be calculated as ( E = \frac{5 \times 12}{2} = 30).

Now, substituting the known values into Euler's formula gives us:

[ V - 30 + 12 = 2 ]

Simplifying this equation:

[ V - 18 = 2 ]

Adding 18 to both sides results in:

[ V = 20 ]

Thus, a dodecahedron has 20 vertices. This is why the correct answer is recognized as the number