Question

1. Verify Euler's formula for each polyhedron:
(b)​

Answer 1

Answer:

Step-by-step explanation:

V - E + F = 2;

or, in words: the number of vertices, minus the number of edges, plus the number of faces, is equal to two.

In the case of the cube, we've already seen that V = 8, E = 12 and F = 6. So,

V - E + F = 8 - 12 + 6 = 14 - 12 = 2

which is what Euler's formula tells us it should be. If we now look at the icosahedron, we find that V = 12, E = 30 and F = 20. Now,

V - E + F = 12 - 30 + 20 = 32 - 30 = 2,

as we expected.

Euler's formula is true for the cube and the icosahedron. It turns out, rather beautifully, that it is true for pretty much every polyhedron. The only polyhedra for which it doesn't work are those that have holes running through them like the one shown in the figure below.

Figure 5: This polyhedron has a hole running through it. Euler's formula does not hold in this case.

Figure 5: This polyhedron has a hole running through it. Euler's formula does not hold in this case.

These polyhedra are called non-simple, in contrast to the ones that don't have holes, which are called simple. Non-simple polyhedra might not be the first to spring to mind, but there are many of them out there, and we can't get away from the fact that Euler's Formula doesn't work for any of them. However, even this awkward fact has become part of a whole new theory about space and shape.