Show that Euler's formula holds for a right pentagonal prism.
Euler's polyhedral formula: V+F-E = 2 V=number of vertices F=number of faces E=number of edges
A right pentagonal prism
V-E+F = 2
10-15+7 = 2
To show that Euler's formula holds for a right pentagonal prism, we need to calculate the number of vertices, edges, and faces of the prism and then verify that the formula holds true.
1. Start by visualizing a right pentagonal prism. It consists of a pentagonal base and a rectangular top, connected by five edges.
2. Let's calculate the number of vertices. The pentagon has 5 vertices, and the rectangular top has 4 vertices. Therefore, the total number of vertices is 5 + 4 = 9.
3. Now, let's calculate the number of edges. The pentagonal base has 5 edges, and the rectangular top has 4 edges. Additionally, each of the five edges connecting the base and the top counts as an edge as well. So, the total number of edges is 5 + 4 + 5 = 14.
4. Finally, let's count the number of faces. The pentagonal base has 1 face, and the rectangular top has 1 face as well. There are also 5 rectangular faces connecting the base and the top. So, the total number of faces is 1 + 1 + 5 = 7.
5. Now, let's apply Euler's formula. According to Euler's formula, the number of vertices (V), edges (E), and faces (F) of a polyhedron are related by the equation V - E + F = 2.
Plugging in the values we calculated, we have:
9 - 14 + 7 = 2
After simplification, we get:
2 = 2
Since both sides of the equation are equal, Euler's formula holds for a right pentagonal prism.
To show that Euler's formula holds for a right pentagonal prism, we need to calculate its Euler characteristic. Euler's formula states that for any convex polyhedron:
V - E + F = 2
where V is the number of vertices, E is the number of edges, and F is the number of faces.
Let's break down the calculation step-by-step:
1. Vertices (V):
In a right pentagonal prism, there are three types of vertices:
- 5 vertices on the top pentagon base
- 5 vertices on the bottom pentagon base
- 5 vertices connecting the corresponding edges of the top and bottom bases
Therefore, the total number of vertices (V) is 5 + 5 + 5 = 15.
2. Edges (E):
In a right pentagonal prism, there are three types of edges:
- 5 edges on the top pentagon base
- 5 edges on the bottom pentagon base
- 10 edges connecting the corresponding vertices of the top and bottom bases
Therefore, the total number of edges (E) is 5 + 5 + 10 = 20.
3. Faces (F):
In a right pentagonal prism, there are three types of faces:
- 2 pentagonal bases
- 5 rectangular lateral faces
Therefore, the total number of faces (F) is 2 + 5 = 7.
Now we can substitute the values into Euler's formula:
V - E + F = 15 - 20 + 7 = 2
Since the result of the calculation is indeed 2, it confirms that Euler's formula holds for a right pentagonal prism.