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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.
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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CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.4You mean for pyramids?

ujjwal
 2 years ago
Best ResponseYou've already chosen the best response.0The base of a cube has 4 sides but it has 4+2 faces.. I didn't get it!

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0Oops, yes, pyramids.

CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.4It seems pretty obvious. There is one face attached to each of the base polygon's sides, plus the base itself.

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah... but I am asking for a solid proof.

ujjwal
 2 years ago
Best ResponseYou've already chosen the best response.0@ParthKohli needs a proof involving serious 'mathematics'

CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.4For a pentagonal prism: dw:1350485379807:dw

CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.4Pfft, drawing pictures is serious enough for me.

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0Oh Lord, a solid proof.

ujjwal
 2 years ago
Best ResponseYou've already chosen the best response.0Well, @CliffSedge , your diagram says it all...

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0I need serious mathematics, sire, serious mathematics.

CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.4Euler's formula might come into play if you want to get all fancy about it. VE+F=2

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0to prove something, you need an equation

ujjwal
 2 years ago
Best ResponseYou've already chosen the best response.0And that formula is valid also for a cube!!

CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.4Euler's formula is valid for all polyhedra, so it will be applicable to the special case of pyramids. Here F=x+1

CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.4The number of vertices, V, also equals x+1 (x for the vertices of the base polygon, plus the top vertex of the pyramid).

CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.4And 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 VE+F=2 F=x+1 V=x+1 E=2x (x+1)2x+(x+1)=2 Simplify and identity is verified.

CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.4Or 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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