## ParthKohli Group Title 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. one year ago one year ago

1. lgbasallote Group Title

there is.

2. ParthKohli Group Title

3. CliffSedge Group Title

You mean for pyramids?

4. ujjwal Group Title

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

5. ParthKohli Group Title

Oops, yes, pyramids.

6. CliffSedge Group Title

It seems pretty obvious. There is one face attached to each of the base polygon's sides, plus the base itself.

7. ParthKohli Group Title

Yeah... but I am asking for a solid proof.

8. ujjwal Group Title

@ParthKohli needs a proof involving serious 'mathematics'

9. CliffSedge Group Title

For a pentagonal prism: |dw:1350485379807:dw|

10. CliffSedge Group Title

Pfft, drawing pictures is serious enough for me.

11. ParthKohli Group Title

Oh Lord, a solid proof.

12. CliffSedge Group Title

LOL, I get it.

13. ParthKohli Group Title

Induction?

14. ujjwal Group Title

Well, @CliffSedge , your diagram says it all...

15. ParthKohli Group Title

I need serious mathematics, sire, serious mathematics.

16. CliffSedge Group Title

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

17. lgbasallote Group Title

to prove something, you need an equation

18. ujjwal Group Title

And that formula is valid also for a cube!!

19. CliffSedge Group Title

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 Group Title

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 Group Title

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 Group Title

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.