How many faces does a pentagonal prism have? Explain. • Describe the faces of a pentagonal prism (different shapes and how many of each shape). • How many edges does a pentagonal prism have? Explain. • How many vertices does a pentagonal prism have? Explain
The Correct Answer and Explanation is:
A pentagonal prism has 7 faces, 15 edges, and 10 vertices.
Explanation
A pentagonal prism is a three-dimensional solid with two congruent, parallel bases that are pentagons and five rectangular lateral faces connecting the corresponding sides of the pentagons.
Faces:
The prism has a total of 7 faces:
- 2 pentagonal faces (top and bottom), which are the bases.
- 5 rectangular faces, each connecting a pair of corresponding sides of the two pentagons.
Each edge of the base pentagon is connected to a rectangle, forming a lateral face. Since a pentagon has five sides, there are five such rectangles. Together with the two pentagonal bases, this gives 2 + 5 = 7 faces.
Edges:
To find the number of edges:
- Each pentagonal base has 5 edges. Since there are two pentagonal bases, that gives 5 × 2 = 10 base edges.
- Each vertex on the bottom pentagon is connected to a corresponding vertex on the top pentagon by a vertical edge. Since a pentagon has 5 vertices, there are 5 vertical edges connecting the two bases.
So, the total number of edges is 5 (bottom) + 5 (top) + 5 (vertical) = 15 edges.
Vertices:
Each pentagonal base has 5 vertices. These are not shared between the bases—they are connected by vertical edges but are distinct points in space. So, 5 (bottom) + 5 (top) = 10 vertices in total.
Summary:
- Faces: 7 (2 pentagons + 5 rectangles)
- Edges: 15 (5 on each base + 5 connecting the bases)
- Vertices: 10 (5 on each base)
The structure of a pentagonal prism follows a general pattern for prisms: 2 polygonal bases and a number of rectangular sides equal to the number of sides in the base polygon
