A polyhedron has 7 faces and 10 vertices. How many edges does the polyhedron have?

Answer

To find the number of edges in a polyhedron with 7 faces and 10 vertices, we can use Euler's formula. Euler's formula states that for any convex polyhedron, the relationship between the number of faces (F), vertices (V), and edges (E) is given by:

\[ F + V = E + 2 \]

### Step-by-Step Solution:

1. **Identify the given values**:
- Number of faces (F) = 7
- Number of vertices (V) = 10

2. **Substitute the values into Euler's formula**:
\[ F + V = E + 2 \]
Substituting the known values:
\[ 7 + 10 = E + 2 \]

3. **Simplify the equation**:
\[ 17 = E + 2 \]

4. **Isolate E (the number of edges)**:
To find E, subtract 2 from both sides:
\[ E = 17 - 2 \]
\[ E = 15 \]

5. **Conclusion**:
The polyhedron has **15 edges**.