If a, b and c are respectively the number of faces, edges and vertices of a pentagonal pyramid, then the value of `((a + b + c)/( 2))` - 12 is

Options

A. `-15`
B. 2
C. 175
D. `-1`

Answer

To solve the problem, we need to determine the values of \( a \), \( b \), and \( c \) for a pentagonal pyramid.

1. **Identify the number of faces (a)**:
A pentagonal pyramid has a pentagonal base and 5 triangular faces that connect the apex to each side of the base. Therefore, the total number of faces is:
\[ a = 1 \text{ (pentagonal base)} + 5 \text{ (triangular faces)} = 6 \]

2. **Identify the number of edges (b)**:
The edges of the pentagonal pyramid consist of the edges of the pentagonal base and the edges connecting the apex to each vertex of the base. The pentagonal base has 5 edges, and there are 5 additional edges connecting the apex to each vertex of the base. Therefore, the total number of edges is:
\[ b = 5 \text{ (base edges)} + 5 \text{ (edges to apex)} = 10 \]

3. **Identify the number of vertices (c)**:
The vertices of the pentagonal pyramid include the 5 vertices of the pentagonal base and 1 apex. Therefore, the total number of vertices is:
\[ c = 5 \text{ (base vertices)} + 1 \text{ (apex)} = 6 \]

4. **Calculate \( a + b + c \)**:
Now we can sum the values of \( a \), \( b \), and \( c \):
\[ a + b + c = 6 + 10 + 6 = 22 \]

5. **Calculate \( \frac{(a + b + c)}{2} - 12 \)**:
Now we need to compute the final expression:
\[ \frac{(a + b + c)}{2} - 12 = \frac{22}{2} - 12 = 11 - 12 = -1 \]

Thus, the final answer is:
\[ \boxed{-1} \]