What is a Polyhedron?

Mary McMahon

A polyhedron is a three dimensional shape bounded by a series of flat planes which are known as faces. A simple example is a cube, a six-faced polyhedron made with square faces. There is actually some dispute in the math world about what constitutes a polyhedron, exactly, and some mathematical disciplines use the term to describe a very specific type of shape. For the most part, however, the definition of a three dimensional shape characterized by having flat faces is reasonably accurate.

Woman standing behind a stack of books
Woman standing behind a stack of books

Each face of a polyhedron is a polygon, another type of geometric figure. Polygons are flat figures made with a series of connecting line segments that form a closed shape. A rectangle is a simple polygon created with four line segments, with a star being a more complex example of a polygon. The polygons in a polyhedron can vary in size, shape, and arrangement considerably, and when all of the polygons are identical, the resulting shape is known as a “symmetrical polyhedron.”

As a general rule, a polyhedron is named according to the number of faces it has. An octahedron has eight faces, a dodecahedron has 12, and so forth. Sometimes, descriptive terms about the shape will be added as well. A pyramid, for example, is a special type of four or five sided polyhedron. Stellated polyhedrons have numerous pyramidal outcroppings called stellations that create a series of points, causing the polyhedron to take on a three-dimensional star shape.

Polyhedra can be convex, which means that a line drawn between any two points on the shape will pass through the shape without straying beyond its boundaries, or nonconvex, in which the line passes outside the shape. A rectangular block is an example of a convex polyhedron: if you draw an imaginary line between any two points on the block, it will pass through the block. By contrast, a stellated polyhedron is nonconvex, because points that go outside the body of the shape can be drawn between the stellations.

People interact with polyhedrons every day, and their study is critical to a wide variety of industries. For example, many products are packaged in polyhedral containers, with the containers being designed for optimum manufacturing efficiency and stackability. Many homes are of these shapes, created with a series of flat planes that have been used to create an enclosed space. The study of polyhedra tends to be introduced to children at a young age, since understanding them can be key to grasping other mathematical concepts.

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Discussion Comments


Please help. As you can tell by my age my 5th grade grandson is doing math that I can't even begin to register. Polyhedron, okay I can read however, my grandson came home with a paper that has on it Name the Polyhedron, a circle? A what looks like pyramid but only 4 solid lines and in the back a dotted line etc. Then it says Name the polyhedron and count its faces, then name and count edges, then count its vertices. I am so lost. There is no math book nothing. Thank you to anyone who can explain this to an old lady so she can understand.

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