## Stretching Gifted Students with an Extension of Euler’s Polyhedra Formula

Anyone who knows anything about Mathematics knows Euler’s Formula connecting the number of faces (F), edges (E) and vertices (V) of polyhedra.

F – E + V = 2

(A cube for example has 6 faces, 12 edges and 8 vertices so that 6 – 12 + 8 = 2).

But what if the shape has a hole in it? (Click the picture to go to the file)

Does the formula change? If so how does it change? And the most important question, why does it change?

Further to this, what if the the shape has 2 holes, 3 holes, …, n holes? How does this affect the formula?

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### 2 Responses to Stretching Gifted Students with an Extension of Euler’s Polyhedra Formula

1. The Dude says:

Hello good sir,

the way you drew the shape, doesn’t the formula hold as is ? Wouldn’t you need to put edges linking the vertices of the “hole” to the vertices of the “original prism”, much as in

for the formula to become
F – E + V = 0
or
F – E + V = 2 – 2n
where n is the number of holes.

I’m maybe missing something.

Good day to you!

• As always, I was so eager to post it that I forgot to put the extra edges in. Thank you kind sir for pointing out the mistake. It has thus been rectified.