For all simple polygons (i.e., without holes), the Euler characteristic equals one. This can be demonstrated for a general figure by the process of triangulation, in which auxiliary lines are drawn connecting vertices so that the region is subdivided into triangles (see figure-->, top). The triangles are then removed one at a time from the outside inward until only one remains, whose Euler characteristic can be easily calculated to equal one. It can be observed that this process of adding and removing lines does not alter the Euler characteristic of the original figure, and so it must also equal one.

For any simple polyhedron (in three dimensions), the Euler characteristic is two, as can be seen by removing one face and “stretching” the remaining figure out onto a plane, resulting in a polygon with a Euler characteristic of one (see figure-->, bottom). Adding the missing face gives a Euler characteristic of two.

For figures with holes, the Euler characteristic will be less by the number of holes present (see figure-->, right), because each hole can be thought of as a “missing” face.