What is the Euler’s formula for a Polyhedron?

Options

A. `F-V-E=0`
B. `F+V-E=2`
C. `F+V+E=1`
D. `F+V+E=3`

Answer

### Step-by-Step Solution:

1. **Understanding Euler's Formula**:
Euler's formula for polyhedra relates the number of faces (F), vertices (V), and edges (E) of a polyhedron. The formula is given by:
\[ F + V - E = 2 \]

2. **Identifying Components**:
- **Faces (F)**: These are the flat surfaces of the polyhedron.
- **Vertices (V)**: These are the corners or points where the edges meet.
- **Edges (E)**: These are the line segments where two faces meet.

3. **Example with a Cube**:
Let's take a cube as an example to illustrate Euler's formula.
- A cube has:
- **Faces (F)**: 6 (top, bottom, left, right, front, back)
- **Vertices (V)**: 8 (corners of the cube)
- **Edges (E)**: 12 (the line segments connecting the vertices)

4. **Substituting Values into the Formula**:
Now, we will substitute the values of F, V, and E into Euler's formula:
\[ F + V - E = 2 \]
Substituting the values:
\[ 6 + 8 - 12 \]

5. **Calculating the Left Side**:
- First, add the number of faces and vertices:
\[ 6 + 8 = 14 \]
- Then, subtract the number of edges:
\[ 14 - 12 = 2 \]

6. **Conclusion**:
Since the left side equals 2, we can confirm that Euler's formula holds true for a cube:
\[ F + V - E = 2 \quad \text{(True)} \]