How many faces, edges, and vertices does a pentagonal prism have? b) How many faces, edges, and vertices does a 100-gonal prism have? c) How many faces, edges, and vertices does an n-gonal prism have?
The Correct Answer and Explanation is:
Let’s analyze each part:
a) Pentagonal Prism
A pentagonal prism has:
- Faces: 7
- 2 pentagonal bases (top and bottom)
- 5 rectangular lateral faces (one for each side of the pentagon)
- Edges: 15
- 5 edges on the top pentagon
- 5 on the bottom pentagon
- 5 vertical edges connecting corresponding vertices of the two pentagons
- Vertices: 10
- 5 on the top pentagon
- 5 on the bottom pentagon
✔ Answer:
Faces: 7, Edges: 15, Vertices: 10
b) 100-gonal Prism
A 100-gonal prism has:
- Faces: 102
- 2 bases (100-gons)
- 100 lateral rectangular faces (one for each side of the base)
- Edges: 300
- 100 edges on top face
- 100 on bottom
- 100 vertical edges
- Vertices: 200
- 100 on the top
- 100 on the bottom
✔ Answer:
Faces: 102, Edges: 300, Vertices: 200
c) n-gonal Prism
For any n-gonal prism:
- Faces: n+2n + 2n+2
- 2 bases (n-gons)
- n rectangular sides
- Edges: 3n3n3n
- n edges per base (top + bottom = 2n)
- n vertical edges
- Vertices: 2n2n2n
- n on top, n on bottom
✔ Answer:
Faces: n+2n + 2n+2, Edges: 3n3n3n, Vertices: 2n2n2n
Explanation:
A prism is a 3D geometric solid formed by two parallel, congruent polygonal bases connected by rectangular lateral faces. In the case of an n-gonal prism, both bases are n-gons (polygons with n sides), and the shape has a uniform cross-section throughout its height.
To calculate the number of faces, consider that every prism has two bases and as many rectangular lateral faces as the number of sides on the base. So, an n-gonal prism has:
- 2 base faces (top and bottom),
- n lateral faces, one for each side of the base,
- Total faces: n+2n + 2n+2.
For edges, each base contributes n edges. Additionally, each vertex on the top connects to its corresponding vertex on the bottom through a vertical edge. So we have:
- 2n base edges (n on each base),
- n vertical edges,
- Total edges: 3n3n3n.
As for vertices, there are n vertices on the top base and n on the bottom, making:
- 2n total vertices.
This formula applies regardless of whether n = 5 (pentagon), 100, or any integer ≥ 3. It gives a consistent and scalable way to understand the structure of prisms.
Understanding prisms through this formulaic lens helps in geometry, architecture, and even computer graphics, where 3D modeling depends on precise geometric descriptions.
