Question

How many faces, edges and vertices of a triangular prism

Show me how you know

Answer 1

A triangular prism is a three-dimensional solid with specific properties related to its faces, edges, and vertices.

### 1. Faces
A triangular prism has:
- **2 triangular faces** (the top and bottom).
- **3 rectangular faces** (the sides connecting the triangles).

So, the total number of faces is:
\[
2 \text{ (triangular)} + 3 \text{ (rectangular)} = 5 \text{ faces}
\]

### 2. Edges
To count the edges, consider:
- Each triangular face has **3 edges**. Since there are **2 triangular faces**, this contributes \(2 \times 3 = 6\).
- Additionally, there are **3 edges** connecting corresponding vertices of the two triangular faces (one edge for each vertex of the triangle).

So, the total number of edges is:
\[
6 \text{ (from the triangles)} + 3 \text{ (connecting edges)} = 9 \text{ edges}
\]

### 3. Vertices
A triangular prism has:
- Each triangular face has **3 vertices**, and since there are **2 triangular faces**, this contributes \(2 \times 3 = 6\) vertices.

So, the total number of vertices is:
\[
6 \text{ vertices}
\]

### Summary
Putting it all together, for a triangular prism:
- **Faces:** 5
- **Edges:** 9
- **Vertices:** 6

This can also be confirmed using Euler's formula for polyhedra, which states:
\[
V - E + F = 2
\]
where \(V\) is the number of vertices, \(E\) is the number of edges, and \(F\) is the number of faces.

Substituting our values:
\[
6 - 9 + 5 = 2
\]
Thus, Euler's formula holds true, confirming our counts are correct.

### Final Count
- **Faces:** 5
- **Edges:** 9
- **Vertices:** 6