Polyhedra are geometric solids limited by polygons, which, in turn, are parts of a plan limited by straight segments that touch each other only at their extremes. You polyhedra they are three-dimensional, so it is possible to observe depth in them, in addition to width and length. Next, we expose and explain the main geometric elements found in polyhedra.
Elements of a polyhedron
all polyhedron has the following elements:
faces: polygons that border the polyhedron;
Edges: straight segments resulting from the meeting of two faces;
vertices: points resulting from the meeting of three or more edges.
convex polyhedra
A plane divides space into two half spaces. This concept is used to define convex polyhedra, which are those that are in the same half-space for every plane that contains one of its faces. In other words, the plane that contains a face of a convex polyhedron it never cuts the other face, leaving part of the polyhedron in one half-space and the other part in another. If that happens, we say that the polyhedron is not convex or concave.
Visually, convex polyhedra does not have concavity. Note the example below: on the left, there is a convex polyhedron; on the right, a non-convex polyhedron.
For convex polyhedra, the Euler relation applies, with some exceptions:
V - A + F = 2
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Polyhedra can be classified according to some of their characteristics. They are usually gathered into three large groups: prisms, pyramids and others. These last ones do not present outstanding characteristics, therefore, they are not discussed.
Prisms
You prisms are polyhedra formed by two congruent and parallel polygonal bases, by the quadrilaterals which connect their corresponding sides and at all points within the region formed by these figures.
The formal definition of prism is as follows: given a polygon A, contained in plane α, and a plane β parallel to plane α, a prism is the geometric solid formed by all line segments whose ends are in polygon A and plane β parallel to a line concurrent to these two plans. The following scheme exemplifies this definition:
Note that every side face of a prism it is a parallelogram.
Pyramids
At pyramids they are polyhedra formed by a polygonal base and triangular side faces that share the “upper vertex”. The following scheme illustrates this definition:
Pyramids whose base is a triangle are called triangular pyramids. Those that have bases formed by quadrilaterals are called quadrangular and so on.
By Luiz Paulo Moreira
Graduated in Mathematics
Would you like to reference this text in a school or academic work? Look:
SILVA, Luiz Paulo Moreira. "What is a polyhedron?"; Brazil School. Available in: https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-poliedro.htm. Accessed on June 27, 2021.
