Question

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Euler's Formula

Find the Verticies for
9 faces: 1 octagon: and 8 triangles

Answer 1

Euler's formula is not needed to answer the question. Consider the first polyhedron. There is only one way to fit the pieces together--it must be a pyramid with a rectangular base. So (picture it!) there are 5 vertices, 8 edges, and 5 faces. Euler's formula says that V-E+F should equal 2, which checks out since 5-8+5 *does* equal 2.

But how could you figure this all out without being able to picture the polyhedron? If you imagine 1 rectangle and 4 triangles made out of paper, then there are 16 edges total, because the triangles contribute 12 edges (3 edges per triangle times 4 triangles) and the rectangle contributes 4 more edges. Every pair of these edges gets glued together to make one edge on the polyhedron, so there are 16/2=8 edges on the polyhedron. Plugging E=8 and F=5 into V-E+F=2 and solving for V gives V=5, as found above.

Consider the second polyhedron. Imagine the paper pieces that you would use to make it. They would have a total of 8+24=32 edges. (8 from the octagon, and 3 for each of the 8 triangles.) So the polyhedron has 32/2=16 edges. We are given that F=9 and just found that E=16. So V-16+9=2, from Euler's formula. Solving for V gives V=2+16-9=9. (This answer makes sense if you picture the second polyhedron!)
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Answer 2

that doesnt give me the answer I looked that up already

Answer 3

try this: http://www.mathsisfun.com/geometry/eulers-formula.html

Answer 4

did it help?