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ParthKohli

  • 2 years ago

I discovered an obvious thing some days back, but is there a solid proof for it? Prove that there are \(\rm x + 1\) faces if the base of a pyramid has \(\rm x\) sides. I know the concept of it, but just need to know if there's a proof.

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  1. lgbasallote
    • 2 years ago
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    there is.

  2. ParthKohli
    • 2 years ago
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    Please.

  3. CliffSedge
    • 2 years ago
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    You mean for pyramids?

  4. ujjwal
    • 2 years ago
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    The base of a cube has 4 sides but it has 4+2 faces.. I didn't get it!

  5. ParthKohli
    • 2 years ago
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    Oops, yes, pyramids.

  6. CliffSedge
    • 2 years ago
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    It seems pretty obvious. There is one face attached to each of the base polygon's sides, plus the base itself.

  7. ParthKohli
    • 2 years ago
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    Yeah... but I am asking for a solid proof.

  8. ujjwal
    • 2 years ago
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    @ParthKohli needs a proof involving serious 'mathematics'

  9. CliffSedge
    • 2 years ago
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    For a pentagonal prism: |dw:1350485379807:dw|

  10. CliffSedge
    • 2 years ago
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    Pfft, drawing pictures is serious enough for me.

  11. ParthKohli
    • 2 years ago
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    Oh Lord, a solid proof.

  12. CliffSedge
    • 2 years ago
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    LOL, I get it.

  13. ParthKohli
    • 2 years ago
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    Induction?

  14. ujjwal
    • 2 years ago
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    Well, @CliffSedge , your diagram says it all...

  15. ParthKohli
    • 2 years ago
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    I need serious mathematics, sire, serious mathematics.

  16. CliffSedge
    • 2 years ago
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    Euler's formula might come into play if you want to get all fancy about it. V-E+F=2

  17. lgbasallote
    • 2 years ago
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    to prove something, you need an equation

  18. ujjwal
    • 2 years ago
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    And that formula is valid also for a cube!!

  19. CliffSedge
    • 2 years ago
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    Euler's formula is valid for all polyhedra, so it will be applicable to the special case of pyramids. Here F=x+1

  20. CliffSedge
    • 2 years ago
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    The number of vertices, V, also equals x+1 (x for the vertices of the base polygon, plus the top vertex of the pyramid).

  21. CliffSedge
    • 2 years ago
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    And the number of edges, E = 2x (x for each side of the base polygon, plus the x edges coming up from its x vertices). These are all based on definitions of pyramids, so V-E+F=2 F=x+1 V=x+1 E=2x (x+1)-2x+(x+1)=2 Simplify and identity is verified.

  22. CliffSedge
    • 2 years ago
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    Or since your task is to show that F=x+1, start with (x+1)-2x+F=2, and solve for F. This all seems quite circular, since it is merely restating definitions of pyramids.

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