Definition of a polyhedron

The elements of a  polyhedron. Euler's polyhedral formula.

The houses we live in, the room we are in now, books, pieces of furniture... almost everything has a polyhedral shape: we live surrounded by polyhedra.

A polyhedron is a solid figure bounded by flat polygonal faces.

Faces are the polygons that form its surface.

Edges are segments, they are the sides of the faces. Each edge borders two faces.

Vertices are the corner points, they are at the ends of the edges. Three or more faces converge in each vertex.

A polyhedron has two space regions, one is inside of it and the other one is outside of it.


You can rotate the figure by moving the red dot.

- What type of polyhedron is it?

- What types of polygons are its base and its lateral faces?

-How many faces does this polyhedron have?

-How many edges does it have?

-How many vertices does it have?

- Type each answer in the corresponding box and hit Enter. When you get a correct answer, write down the values in your notebook.

- Click on Init to get a different polyhedron (prism or pyramid). Repeat the previous activity and elaborate a table like this one with the data from several polyhedra.

faces + vertices

- Observe the data in this table and find out the relation between the numbers in the two columns on the right.


The mathematician Euler proved the relation among the numbers of faces, the numbers of vertices and the numbers of edges of a non toroidal polyhedron.

A non toroidal polyhedron is understood as the one that doesn't form a ring, neither do any of its faces or edges. Those polyhedra - as they satisfy Euler's formula - are called Eulerian.

  Eduardo Barbero Corral
Spanish Ministry of Education , Social Afairs and Sport. Year 2007

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