Euler's formula holds for every solid figure consisting of
vertices, flat faces, and straight edges.

For any solid figure with only flat surfaces and 
straight edges, this formula always holds:

Number of vertices + number of faces = number of edges + 2

Here is a demonstration of that formula with a "box",
that is, a rectangular solid:




Count the vertices (sharp corners).  They are 

A, B, C, D, E, F, G, H

That's 8.  So for the above figure, v = 8

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Count the faces (flat surfaces).  They are rectangles 

ABCD, EFGH, AEHD, BFGC, ABFE, DCGH, 

That's 6.  So for the above figure, f = 6

-----
  
Count the edges.  They are line segments: 

AB, BC, CD, DA, EF, FG, GH, HE, AE, BF, DH, CG

That's 12.  So for the above figure, e = 12

-----

Now look at Euler's formula:



Substituting,





So you see that Euler's formula holds for the
above rectangular solid.

Euler's formula holds for every solid figure consisting of
vertices, flat faces, and straight edges.

Edwin