Polyhedra: Definition & Types

Instructor: Chris Malec

Chris has a PhD in Physics

This article defines the meaning of polyhedra, a type of geometric object. Several types of polyhedra are described, and some interesting properties of polyhedra are discussed.

Roll of the Dice

Have you ever played a game with something called a polyhedra? Chances are you have, even if you've never heard the term. If you've ever played Yahtzee, Boggle or any other game involving a pair of six-sided dice, you've played with a polyhedra. That's because a six-sided dice is actually a type of polyhedra called a cube. When you look around, you'll notice there are actually lots of polyhedra in your everyday life, many of them sided, like appliances and boxes.

What are Polyhedra?

Polyhedra are very general shapes. Even shapes that are not polyhedra can be simulated by polyhedra; in fact, this is what happens in many computer graphics. So what are they? Polyhedra are three-dimensional objects made of polygons. Polygons are two-dimensional objects made of straight lines; they can be squares, rectangles, triangles, nonagons, etc.

There are three vocabulary words that are important to remember in order to understand what polyhedra are: faces, edges, and vertices. Faces are the flat parts of the polyhedra -- they are the polygons. Edges are are the line where two faces meet. Vertices are the points where three or more faces meet.

A cube, displaying face, vertex, and edge.

In fact, there is a mathematical relationship between these three terms, known as Euler's Polyhedron Formula. It states that the vertices, edges, and faces satisfy the equation: Vertices - Edges + Faces = 2.

Types of Polyhedra

So the only requirements for a shape to be a polyhedron is that all the faces of the polyhedron are polygons and that all the edges of all the faces fit together so there are no gaps. Below is an example of a regular polyhedron; this one is called a dodecahedron.

An example of a polyhedron, this one is called a dodecahedron.

A dodecahedron is a regular polyhedron where all the faces are regular pentagons. Polyhedra don't have to be regular, meaning that all the faces are the same, and all the polygons are regular.

Image created by Robert Webb

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