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.

1. lgbasallote

there is.

2. ParthKohli

3. CliffSedge

You mean for pyramids?

4. ujjwal

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

5. ParthKohli

Oops, yes, pyramids.

6. CliffSedge

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

7. ParthKohli

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

8. ujjwal

@ParthKohli needs a proof involving serious 'mathematics'

9. CliffSedge

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

10. CliffSedge

Pfft, drawing pictures is serious enough for me.

11. ParthKohli

Oh Lord, a solid proof.

12. CliffSedge

LOL, I get it.

13. ParthKohli

Induction?

14. ujjwal

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

15. ParthKohli

I need serious mathematics, sire, serious mathematics.

16. CliffSedge

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

17. lgbasallote

to prove something, you need an equation

18. ujjwal

And that formula is valid also for a cube!!

19. CliffSedge

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

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

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

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.