Question:
A right prism having each of the bases a polygon of n sides.
- Number of faces = 2 polygons of n sides + n rectangles = (n+2).
- Number of edges = 3n.
- Number of vertices = 2n.
- Verify Euler’s formula: \( F - E + V = 2 \).
Explanation:
1. Number of Faces (F):
- The prism has two bases, each an n-sided polygon.
- Each side of the polygon base has a corresponding rectangular face connecting to the other base, so there are n rectangular faces.
- Therefore, total faces \( = n + 2 \).
2. Number of Edges (E):
- Each polygonal base has n edges.
- There are n vertical edges connecting corresponding vertices of the two bases.
- Therefore, total edges \( = 3n \).
3. Number of Vertices (V):
- Each base has n vertices.
- Since there are two bases, the total number of vertices \( = 2n \).
Euler’s Formula:
Euler’s formula states \( F - E + V = 2 \).
Verify this:
- \( F = n + 2 \)
- \( E = 3n \)
- \( V = 2n \)
Substituting these into Euler’s formula:
\[ (n + 2) - 3n + 2n = 2 \]
Simplifying:
\[ n + 2 - 3n + 2n = 2 \]
\[ 2 = 2 \]
This confirms Euler’s formula holds true.
I hope this explanation helps you understand the structure and properties of a right prism with an n-sided polygon base easily.
