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 go through each part step by step.
a) Pentagonal Prism:
A pentagonal prism is a 3D shape formed by joining two pentagonal faces and connecting them with rectangular faces.
- Faces:
A pentagonal prism has 7 faces. There are 2 pentagonal faces (one at the top and one at the bottom) and 5 rectangular faces connecting the sides of the pentagons. - Edges:
It has 15 edges. Each pentagonal face has 5 edges, and there are 5 edges connecting the corresponding vertices of the two pentagons (the vertical edges). So the total number of edges is 5+5+5=155 + 5 + 5 = 155+5+5=15. - Vertices:
A pentagonal prism has 10 vertices. Each pentagon has 5 vertices, and since there are two pentagons, the total number of vertices is 5×2=105 \times 2 = 105×2=10.
b) 100-gonal Prism:
A 100-gonal prism is a 3D shape with 100 sides in the base, where the base is a 100-sided polygon.
- Faces:
It has 102 faces. There are 2 polygonal faces (one at the top and one at the bottom), and 100 rectangular faces connecting the corresponding edges of the two 100-gons. - Edges:
A 100-gonal prism has 300 edges. Each 100-gon has 100 edges, and there are 100 edges connecting the corresponding vertices between the two 100-gons. The total number of edges is 100+100+100=300100 + 100 + 100 = 300100+100+100=300. - Vertices:
It has 200 vertices. Each 100-gon has 100 vertices, and there are two 100-gons, so the total number of vertices is 100×2=200100 \times 2 = 200100×2=200.
c) n-gonal Prism:
An n-gonal prism is a 3D shape with an n-sided polygon as the base.
- Faces:
An n-gonal prism has n+2n + 2n+2 faces. There are 2 polygonal faces (the top and bottom) and nnn rectangular faces connecting the corresponding edges of the two n-gons. - Edges:
The total number of edges in an n-gonal prism is 3n3n3n. Each n-gon has n edges, and there are n edges connecting the corresponding vertices of the two n-gons, so the total number of edges is n+n+n=3nn + n + n = 3nn+n+n=3n. - Vertices:
The number of vertices in an n-gonal prism is 2n2n2n. Each n-gon has n vertices, and there are two n-gons, so the total number of vertices is n×2=2nn \times 2 = 2nn×2=2n.
Summary:
- Pentagonal Prism:
- Faces: 7
- Edges: 15
- Vertices: 10
- 100-gonal Prism:
- Faces: 102
- Edges: 300
- Vertices: 200
- n-gonal Prism:
- Faces: n+2n + 2n+2
- Edges: 3n3n3n
- Vertices: 2n2n2n
This pattern holds for any n-gonal prism, where the number of faces, edges, and vertices depends on the number of sides (n) of the polygon forming the base.
