determine why Euler's formula continues to hold for the polyhedron formed by cutting corners off of an octahedron.
a) for an octahedron V=__, F=__,E=__
b) when you slice off one corner of the octahedron, you (gain or lose)__ edges.
C) therefore, the total change in V is __, the total change in F is __, and the total change in V+F is ___.
I have for a v=6, F=8 and E=12 but that does not come out to showing Euler formula correctly.
For an octahedron, V =__8__, F = __6__, and E =
__12___.
To determine why Euler's formula continues to hold for the polyhedron formed by cutting corners off of an octahedron, let's break it down step by step:
a) For an octahedron, we have:
- V (number of vertices) = 6 (since an octahedron has 6 vertices)
- F (number of faces) = 8 (an octahedron has 8 faces)
- E (number of edges) = 12 (an octahedron has 12 edges)
Your values for V, F, and E are correct for an octahedron.
b) When you slice off one corner of the octahedron, you would lose three edges. This is because each corner shares three edges with the surrounding faces, and by cutting off one corner, you remove those three edges connected to it.
c) Now, let's calculate the total change in V, F, and E after slicing off one corner:
- Change in V: When slicing off one corner, we remove one vertex, so the change in V is -1.
- Change in F: We also remove one face when slicing off one corner, so the change in F is -1.
- Change in E: As mentioned earlier, we lose three edges when slicing off one corner, so the change in E is -3.
Now, we can calculate the total change in V+F:
Total change in V+F = change in V + change in F
Total change in V+F = -1 + (-1)
Total change in V+F = -2
Since the change in V+F is -2, this means that after slicing off one corner of the octahedron, the resulting polyhedron has 2 fewer vertices and 2 fewer faces.
To check if Euler's formula still holds, let's plug the new values into Euler's formula:
Vertices + Faces - Edges = 0
New V + New F - New E = 0
(6-1) + (8-1) - (12-3) = 0
5 + 7 - 9 = 0
Therefore, Euler's formula still holds for the polyhedron formed by cutting corners off of an octahedron, as the equation evaluates to 0.
I hope this explanation clarifies why Euler's formula continues to hold for the modified polyhedron!