A polyhedron is a three-dimensional shape made entirely of flat faces, where every face is a polygon. Think of a cube, a pyramid, or a diamond shape: each one is built from flat surfaces (squares, triangles, pentagons) that connect along straight edges and meet at pointed corners. If any surface is curved, like on a sphere or cylinder, it’s not a polyhedron.
Faces, Edges, and Vertices
Every polyhedron is described by three basic parts. The flat polygon surfaces are called faces. The line segments where two faces meet are called edges. The points where edges come together are called vertices (a single one is a vertex). A cube, for example, has 6 faces, 12 edges, and 8 vertices.
These three numbers aren’t independent of each other. For any convex polyhedron (one with no indentations or holes), the number of vertices minus the number of edges plus the number of faces always equals 2. Written out: V − E + F = 2. This relationship, known as Euler’s formula, holds universally. A tetrahedron has 4 vertices, 6 edges, and 4 faces: 4 − 6 + 4 = 2. You can verify it with any convex polyhedron you encounter, and it always works.
Prisms and Pyramids
The two most common families of polyhedra are prisms and pyramids. A prism has two identical polygon bases sitting in parallel planes, connected by rectangular faces. A triangular prism (like a Toblerone box) has two triangular bases and three rectangles. A rectangular prism is a box. Every cross-section you cut parallel to the base has the same area, which is what makes a prism a prism.
A pyramid has a single polygon base and an apex point above it, with triangular faces connecting each edge of the base to that apex. A square pyramid, like the ones in Egypt, has a square base and four triangular faces. A right pyramid has its apex centered directly above the base, while an oblique pyramid has the apex shifted to one side. Both are still polyhedra because every surface remains a flat polygon.
Regular Polyhedra: The Platonic Solids
A polyhedron is considered regular when all of its faces are identical regular polygons (equal sides, equal angles) and the same number of faces meet at every vertex. This is a surprisingly strict requirement. A square-based pyramid has regular faces (equilateral triangles and a square), but it doesn’t qualify because the faces aren’t all the same type.
Only five shapes in all of geometry meet both criteria. These are called the Platonic solids:
- Tetrahedron: 4 triangular faces
- Cube: 6 square faces
- Octahedron: 8 triangular faces
- Dodecahedron: 12 pentagonal faces
- Icosahedron: 20 triangular faces
That’s it. No matter how creative you get, you cannot construct a sixth regular polyhedron. The geometry simply doesn’t allow it. The reason comes down to angles: the interior angles of the polygons meeting at each vertex must add up to less than 360 degrees, or the shape lies flat instead of curving into three dimensions. Only five combinations of regular polygons satisfy that constraint. If you try to tile a flat surface with three squares meeting at each corner, the pattern naturally curls up into the third dimension and becomes a cube.
Semiregular Polyhedra
Relaxing the rules slightly opens up another family. If you still require that all faces be regular polygons and that every vertex look the same, but allow more than one type of polygon, you get the Archimedean solids. There are exactly 13 of them. A soccer ball is the most familiar example: its surface is a truncated icosahedron, made of 12 pentagons and 20 hexagons, with the same arrangement of shapes around every vertex.
The key distinction from the Platonic solids is that Archimedean solids use two or more types of regular polygon, while Platonic solids use only one. Both families share extremely high symmetry, which is why they look so visually balanced from every angle.
What Doesn’t Count as a Polyhedron
Shapes with any curved surface are not polyhedra. Cylinders, cones, and spheres all fail the test because their surfaces aren’t flat polygons. A hemisphere sitting on a circular base isn’t a polyhedron either, since circles aren’t polygons.
It’s also worth noting the distinction between convex and concave polyhedra. A convex polyhedron has no indentations: if you draw a line between any two points inside it, the line stays inside. Star-shaped polyhedra and polyhedra with “dents” are concave. Both are valid polyhedra, but Euler’s formula (V − E + F = 2) applies cleanly to the convex ones. Concave polyhedra with holes or tunnels through them can produce different values.
Polyhedra in the Physical World
Polyhedral shapes show up constantly in nature, especially in crystals. The internal structure of a crystal is built from repeating units that can be described as space-filling polyhedra, primarily tetrahedra and octahedra. In a face-centered cubic crystal lattice (the structure found in metals like aluminum, copper, and gold), the smallest repeating unit is made of two tetrahedra and one octahedron packed together. A body-centered cubic lattice, found in iron and chromium, is built entirely from tetrahedra. These geometric arrangements directly influence a material’s properties, including how well it conducts electricity and how ions move through it.
Dice are polyhedra, too. Standard six-sided dice are cubes. The 20-sided dice used in tabletop games are icosahedra. Geodesic domes in architecture approximate spheres using triangular faces, making them polyhedral structures. Even the molecular geometry of certain viruses follows an icosahedral pattern, packing proteins efficiently into a roughly spherical shell using flat, repeating faces.