a) How many faces does a prism with a pentagonal base have? What are the shapes of the faces? b) How many edges does a prism with a pentagonal base have? c) How many vertices does a prism with a pentagonal base have?
The Correct Answer and Explanation is:
Let’s break down each part of the question for a prism with a pentagonal base.
a) How many faces does a prism with a pentagonal base have? What are the shapes of the faces?
A prism is defined as a polyhedron with two congruent polygonal bases and rectangular lateral faces. For a pentagonal prism, the base is a pentagon. The number of faces in a prism is given by the sum of the two bases and the lateral faces.
- There are 2 pentagonal faces (one at the top and one at the bottom).
- There are 5 rectangular faces that connect the sides of the pentagon bases.
So, the total number of faces in a pentagonal prism is:
2 pentagonal faces + 5 rectangular faces = 7 faces.
b) How many edges does a prism with a pentagonal base have?
The number of edges in a prism can be calculated by counting the edges of both bases and the lateral edges that connect them.
- Each pentagonal base has 5 edges.
- There are 5 lateral edges that connect the corresponding vertices of the two pentagonal bases.
So, the total number of edges in a pentagonal prism is:
5 edges on the top base + 5 edges on the bottom base + 5 lateral edges = 15 edges.
c) How many vertices does a prism with a pentagonal base have?
Each pentagonal base has 5 vertices, and since there are two bases (top and bottom), the prism will have:
5 vertices on the top base + 5 vertices on the bottom base = 10 vertices.
Summary:
- Faces: 7 (2 pentagonal faces and 5 rectangular faces)
- Edges: 15 (5 edges on each of the two bases and 5 lateral edges)
- Vertices: 10 (5 vertices on each of the two bases)
