A prism has 96 edges. How many vertices and faces does it have? Justify your answer.
The Correct Answer and Explanation is:
Correct Answer:
The prism has 48 vertices and 50 faces.
Explanation:
To solve this problem, let us recall the general properties of a prism:
A prism is a three-dimensional solid with two congruent, parallel polygonal bases and rectangular lateral faces connecting the corresponding edges of the bases. The number of edges, vertices, and faces depends on the number of sides of the polygon that forms the base.
Let:
- n = number of sides of the polygonal base
- E = total number of edges
- V = total number of vertices
- F = total number of faces
For a prism with an n-sided polygon as its base:
- Number of vertices (V) = 2 × n
- Number of edges (E) = 3 × n
- Number of faces (F) = n + 2
The reason is as follows:
- Each base contributes n edges, and there are n lateral edges connecting the corresponding vertices of the two bases, making a total of 3n edges
- The two bases each have n vertices, so the total number of vertices is 2n
- There are n lateral faces, plus the two polygonal bases, making n + 2 faces
Step-by-step Calculation:
We are told that the prism has 96 edges, which follows the formula:E=3n=96E = 3n = 96E=3n=96
Solving for n:n=963=32n = \frac{96}{3} = 32n=396=32
Therefore:
- Vertices (V) = 2 × n = 2 × 32 = 64
- Edges (E) = 3 × n = 3 × 32 = 96 (which matches the given value)
- Faces (F) = n + 2 = 32 + 2 = 34
Thus, based on the standard prism formulas, if there are 96 edges, the prism has:
- 64 vertices
- 34 faces
There seems to be a miscalculation in the earlier provided answer. Correct final answer is:
Vertices = 64
Faces = 34
The formulas come from the geometric structure of prisms. Each additional side on the polygonal base adds one more vertex per base, three more edges, and one more face. Using these relations ensures consistent results for any polygonal prism.
