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Polygons and PolyhedraPolyhedra

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Up to now we have just looked at what we can do with polygons in a flat, two-dimensional world. A polyhedron is a 3-dimensional object that is made up of polygons. Here are some examples:

Polyhedra cannot contain curved surfaces – spheres and cylinders, for example, are not polyhedra.

The polygons that make up a polyhedron are called its faces. The lines where two faces are connected are called edges, and the corners where the edges meet are called vertices.

Polyhedra come in many different shapes and sizes – from simple cubes or pyramids with just a few faces, to complex objects like the star above, which has 60 triangular faces. It turns out, however, that all polyhedra have one important property in common:

Euler’s Polyhedron Formula
In every polyhedron, the number of faces (F) plus the number of vertices (V) is two more than the number of edges (E). In other words,

F+V=E+2

For example, if a polyhedron has 12 faces and 18 vertices, we know that it must have edges.

This equation was discovered by the famous Swiss mathematician Leonard Euler. It is true for any polyhedron, as long as it doesn’t contain any holes.

If you try different polyhedra, like the ones above, you’ll find that Euler’s formula always works. In a later course you’ll learn how to actually prove it mathematically.