A polyhedron has 30 edges and 12 surfaces. How many vertices of this polyhedron?

Options

A. 12
B. 15
C. 20
D. 24

Answer

To find the number of vertices (V) of the polyhedron, we can use Euler's formula, which states:

\[ V - E + F = 2 \]

where:
- \( V \) = number of vertices
- \( E \) = number of edges
- \( F \) = number of faces

Given:
- \( E = 30 \) (number of edges)
- \( F = 12 \) (number of faces)

We need to find \( V \).

### Step-by-Step Solution:

1. **Write down Euler's formula:**
\[ V - E + F = 2 \]

2. **Substitute the known values into the formula:**
\[ V - 30 + 12 = 2 \]

3. **Simplify the equation:**
\[ V - 18 = 2 \]

4. **Solve for \( V \):**
\[ V = 2 + 18 \]
\[ V = 20 \]

Thus, the number of vertices \( V \) of the polyhedron is **20**.

### Final Answer:
The number of vertices of the polyhedron is **20**.