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Polyhedrons

On three-dimensional geometric objects with straight faces and edges: polyhedrons

Polyhedron means solid with multiple faces. These shapes are bounded exclusively by polygons and possess no curvature anywhere. The number of possible polyhedra is infinite. They can be classified into various categories based on their properties. One important property is whether a solid is convex or non-convex. Non-convex polyhedra feature concave edge angles, re-entrant regions, that is areas directed inwards. Examples include star polyhedra, such as the stellated octahedron. Another classification criterion is the regularity of these shapes. Convex solids exhibiting complete regularity are the Platonic solids, of which there are exactly five. The 13 Archimedean solids and the Catalan solids derived from them possess a lesser degree of regularity. These form their own distinct categories, as do the Johnson solids. Like the Platonic and Archimedean solids, the latter are bounded by regular polygons. This also applies to the prisms and antiprisms within this category. The simplest shape on this list is the cuboid, which is a special case of both a prism and a rhombohedron. The rhombohedron, in turn, is a special case of a parallelepiped. Pyramids, particularly the regular square pyramid, are also well-known polyhedral shapes. Generally, calculations involving polyhedra become increasingly complicated as the number of different faces and non-right dihedral angles rises. An example here is the general tetrahedron. Despite having only four irregular faces, it requires lengthy formulas to calculate its surface area and volume.
Euler's formula applies to all convex polyhedra: the number of vertices plus the number of faces minus the number of edges always equals two, V+F−E=2.

The following 56 polyhedrons can be calculated on these pages:

Last updated on 08/12/2026. Author: Jürgen Kummer

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