How many faces and edges does a triangular pyramid have ? What is the other name of the triangular pyramid ?

Answer

To solve the question about how many faces and edges a triangular pyramid has, and to find its other name, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure of a Triangular Pyramid**:
- A triangular pyramid, also known as a tetrahedron, has a triangular base and three triangular faces that connect the base to a single apex (top point).

2. **Counting the Faces**:
- The triangular pyramid has:
- 1 triangular base (the bottom face).
- 3 triangular lateral faces (the sides).
- Therefore, the total number of faces is:
\[ \text{Total Faces} = 1 + 3 = 4 \]

3. **Counting the Edges**:
- The edges of a triangular pyramid are the line segments where two faces meet.
- The edges can be counted as follows:
- 3 edges from the base triangle.
- 3 edges connecting the apex to each vertex of the base triangle.
- Therefore, the total number of edges is:
\[ \text{Total Edges} = 3 + 3 = 6 \]

4. **Counting the Vertices**:
- The vertices of a triangular pyramid are the points where the edges meet.
- There are:
- 3 vertices at the base.
- 1 vertex at the apex.
- Therefore, the total number of vertices is:
\[ \text{Total Vertices} = 3 + 1 = 4 \]

5. **Verifying with Euler's Formula**:
- Euler's formula states that for any convex polyhedron:
\[ V - E + F = 2 \]
where \( V \) is the number of vertices, \( E \) is the number of edges, and \( F \) is the number of faces.
- Substituting the values we found:
\[ 4 - 6 + 4 = 2 \]
- This confirms our counts are correct.

6. **Identifying the Other Name**:
- The other name for a triangular pyramid is a **tetrahedron**.

### Final Answer:
- A triangular pyramid has **4 faces** and **6 edges**.
- The other name for a triangular pyramid is **tetrahedron**.