SOLUTION: Use Euler’s Formula to find the number of vertices in the polyhedron with 11 faces, 1 decagon, and 10 triangles. a. 11 b. 22 c. 6 d. 10

Algebra ->  Polygons -> SOLUTION: Use Euler’s Formula to find the number of vertices in the polyhedron with 11 faces, 1 decagon, and 10 triangles. a. 11 b. 22 c. 6 d. 10      Log On


   

Question 1141737: Use Euler’s Formula to find the number of vertices in the polyhedron with 11 faces, 1 decagon, and 10 triangles.
a. 11
b. 22
c. 6
d. 10
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.
The Euler's formula is


    F - E + V = 2


    (faces - edges + vertices = 2).


Substitute  F = 11, E = 10 + 10 = 20 into the formula


    11 - 20 + V = 2,


and you will get


    V = 2 - 11 + 20 = 11 vertices.      ANSWER


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Euler's Formula: F - E + V = 2.

The polyhedron has 11 faces -- that is given.

The polyhedron has one decagon face and 10 triangle faces. That is 10+10(3) = 40 edges. An edge of the polyhedron is where 2 polygon edges meet; so the number of edges of the polyhedron is 40/2 = 20.

Then

11+-+20+%2B+V+=+2
V+=+11

The polyhedron has 11 vertices.

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While Euler's Formula is useful for solving many problems, it is not needed here. A little thinking shows that the polyhedron is a pyramid with a decagon for a base. The base has 10 vertices; the peak of the pyramid is the only other vertex.

10+1 = 11 vertices.