Which solid has sum of faces and vertices as 10 ?

Options

A. Cuboid
B. Cone
C. Square pyramid
D. Triangular prism

Answer

To solve the problem of finding which solid has the sum of faces and vertices equal to 10, we can use Euler's formula, which states:

\[ f + v = e + 2 \]

Where:
- \( f \) = number of faces
- \( v \) = number of vertices
- \( e \) = number of edges

Given that the sum of faces and vertices \( f + v = 10 \), we can rearrange Euler's formula to find the number of edges:

1. Start with Euler's formula:
\[ f + v = e + 2 \]

2. Substitute \( f + v = 10 \) into the formula:
\[ 10 = e + 2 \]

3. Rearranging gives us:
\[ e = 10 - 2 \]
\[ e = 8 \]

Now, we need to find a solid that has 8 edges, and we will check some common solids:

1. **Cuboid**:
- Faces: 6
- Vertices: 8
- Edges: 12
- Sum of faces and vertices: \( 6 + 8 = 14 \) (not suitable)

2. **Cone**:
- Faces: 2 (1 circular base + 1 curved surface)
- Vertices: 1 (the apex)
- Edges: 0
- Sum of faces and vertices: \( 2 + 1 = 3 \) (not suitable)

3. **Square Pyramid**:
- Faces: 5 (1 square base + 4 triangular faces)
- Vertices: 5 (4 at the base + 1 apex)
- Edges: 8 (4 base edges + 4 edges connecting the apex to the base)
- Sum of faces and vertices: \( 5 + 5 = 10 \) (suitable)

4. **Triangular Prism**:
- Faces: 5 (2 triangular bases + 3 rectangular faces)
- Vertices: 6 (3 vertices from each triangle)
- Edges: 9 (3 edges from each triangle + 3 connecting edges)
- Sum of faces and vertices: \( 5 + 6 = 11 \) (not suitable)

From the analysis, the solid that has the sum of faces and vertices equal to 10 is the **Square Pyramid**.

### Summary of Steps:
1. Write down Euler's formula.
2. Substitute the given sum of faces and vertices into the formula.
3. Rearrange to find the number of edges.
4. Check common solids to find one that matches the criteria.