Consider a pyramid whose base is a regular n-gon—that is, a regular polygon with n sides. How many vertices would such a pyramid have? How many faces? How many edges? (a) Describe the relationship between the number of vertices of an n-gon prism and of an n-gon pyramid. (Select one) An n-gon prism has n + 1 more vertices than an n-gon pyramid. An n-gon prism has n − 2 more vertices than an n-gon pyramid. An n-gon prism has n more vertices than an n-gon pyramid. An n-gon prism has n − 1 more vertices than an n-gon pyramid. An n-gon prism has n + 2 more vertices than an n-gon pyramid. (b) Describe the relationship between the number of faces of an n-gon prism and of an n-gon pyramid. (Select one) An n-gon prism has one more face than an n-gon pyramid. An n-gon prism has two fewer faces than an n-gon pyramid. An n-gon prism has two more faces than an n-gon pyramid. An n-gon prism has one less face than an n-gon pyramid. An n-gon prism has the same number of faces as an n-gon pyramid. (c) Describe the relationship between the number of edges of an n-gon prism and of an n-gon pyramid. (Select one) An n-gon prism has n − 1 more edges than an n-gon pyramid. An n-gon prism has n − 2 more edges than an n-gon pyramid. An n-gon prism has n more edges than an n-gon pyramid. An n-gon prism has n + 2 more edges than an n-gon pyramid. An n-gon prism has n + 1 more edges than an n-gon pyramid
The Correct Answer and Explanation is:
To solve this problem, we need to understand the properties of an n-gon pyramid and an n-gon prism. Let’s break it down carefully:
n-gon Pyramid
- A pyramid with a regular n-gon base has:
- Vertices: nnn base vertices plus 1 apex = n+1n + 1n+1 vertices
- Faces: nnn triangular lateral faces plus 1 base = n+1n + 1n+1 faces
- Edges: nnn base edges plus nnn edges from the apex to each base vertex = 2n2n2n edges
n-gon Prism
- A prism has two congruent n-gon bases and vertical edges connecting corresponding vertices:
- Vertices: 2n2n2n (n from top base and n from bottom base)
- Faces: nnn rectangular lateral faces plus 2 n-gon bases = n+2n + 2n+2 faces
- Edges: nnn top base edges + nnn bottom base edges + nnn vertical edges = 3n3n3n edges
(a) Relationship Between Vertices
- Pyramid: n+1n + 1n+1 vertices
- Prism: 2n2n2n vertices
- Difference: 2n−(n+1)=n−12n – (n + 1) = n – 12n−(n+1)=n−1
✅ Correct answer: An n-gon prism has n − 1 more vertices than an n-gon pyramid.
(b) Relationship Between Faces
- Pyramid: n+1n + 1n+1 faces
- Prism: n+2n + 2n+2 faces
- Difference: (n+2)−(n+1)=1(n + 2) – (n + 1) = 1(n+2)−(n+1)=1
✅ Correct answer: An n-gon prism has one more face than an n-gon pyramid.
(c) Relationship Between Edges
- Pyramid: 2n2n2n edges
- Prism: 3n3n3n edges
- Difference: 3n−2n=n3n – 2n = n3n−2n=n
✅ Correct answer: An n-gon prism has n more edges than an n-gon pyramid.
Summary:
- (a): An n-gon prism has n − 1 more vertices than an n-gon pyramid.
- (b): An n-gon prism has one more face than an n-gon pyramid.
- (c): An n-gon prism has n more edges than an n-gon pyramid.
These relationships hold true for any polygon with n sides and illustrate how adding a second base (as in a prism) changes the geometric structure.
