Answers:
- faces = n+2
- edges = 3n
- vertices = 2n
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Explanation:
Think of a hexagonal room with n = 6 walls, i.e. the floor is a hexagon with n = 6 sides. The floor and ceiling are parallel to each other, and congruent hexagons. That's 2 faces so far. Then we have another 6 faces to account for the walls. This gives 2+6 = 8 faces of a hexagonal prism.
In more general terms, a prism with a base of n sides will have 2 parallel and congruent base faces, and n walls or lateral faces. This gives n+2 total faces.
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Let's go back to the hexagonal prism. The floor has 6 sides to it, and so does the ceiling. We have 6+6 = 12 edges so far. Then we have another 6 edges where each of the rectangular walls meet up. That gives 12+6 = 18 edges total of this hexagonal prism room.
For any more general case, each base has n sides. That gives 2n sides so far for just the bases. Then add on another n for the lateral edges and we get 3n total edges.
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Once again lets revisit the room with the hexagonal floor and ceiling. The floor has 6 vertices and the ceiling has the same vertex count. Therefore, this prism has 6+6 = 12 vertices.
For the general case, each base has n vertices. There are 2 such identical bases giving 2n vertices total.
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One way to check the answer:
We could use Euler's Polyhedron Formula which is
F+V-E = 2
where,
- F = number of faces
- V = number of vertices
- E = number of edges
For the hexagonal prism we found
Then notice how
F+V-E = 2
8+12-18 = 2
20-18 = 2
2 = 2
This confirms the formula works for a hexagonal prism.
Now let's check it for the more general case
We found earlier that,
So,
F+V-E = 2
n+2+2n-3n = 2
3n-3n+2 = 2
0n+2 = 2
0+2 = 2
2 = 2
This helps confirm the answer for any prism with the base of n sides.
