A poly-*what*? In geometry classes for young learners, we focus on one-dimensional, two-dimensional, and three-dimensional shapes. They all have special properties, and polyhedrons are no different. A polyhedron is a three-dimensional shape with flat polygonal surfaces, straight edges and sharp corners (vertices). It consists of a group of polygons all connected at the edges. Some common examples include cubes, prisms, and pyramids.

Some other common three-dimensional shapes, such as cones, spheres, and cylinders, however, are not polyhedrons. They don’t have polygonal surfaces, meaning they do not have corners or edges. And in case you’re wondering, the plural of polyhedron is polyhedra or polyhedrons. Let’s take a look at what polyhedrons consist of, the different shapes and types, as well as the formula to use when looking at the relationship between the faces, edges, and vertices in a polyhedron.

## Faces, Edges, and Vertices

There are three important components of a polyhedron: faces, edges, and vertices. A face is a two-dimensional flat surface on a polyhedron, basically a polygon. An edge of a polyhedron is a line segment where two faces meet. Where two edges meet is called the vertex (or vertices for plural). Let’s use a box as an example. One flat side of a box would be called a face, while an edge of the box from one corner to the other would be called the edge, and a corner of the box would be the vertex. Now that we understand what the components of a polyhedron are, let’s get into the different shapes that a polyhedron can take.

## Shapes

The word “polyhedron” comes from the Greek words *poly *and *hedron*. “Poly” meaning many, and “hedron” meaning surface. The name of a polyhedron is dependent on the amount of faces it has. A polyhedron with 4 faces would be called a tetrahedron. A polyhedron with 5 faces would be a pentahedron, and so on. Some of these shapes are given simpler names like prism or cube, so as to lessen the confusion between them. Examples of polyhedrons include a diamond, a pyramid, a Rubik’s Cube, or a soccer ball

## Classifications and Types

Polyhedrons can generally be classified into three different categories: prisms, pyramids, and platonic solids. Prisms and pyramids are considered irregular polyhedrons, while platonic solids are regular polyhedrons.

### Irregular Polyhedrons

Irregular polyhedrons are those in which the faces are not congruent (or identical) to each other. A prism is a shape of which both ends are identical polygons with flat side faces (rectangles or parallelograms). A prism is named after its base, which can be a square, rectangle, triangle, or any *n*-sided polygon. A great example of this is a Toblerone chocolate bar: both ends are identical triangles, and the side faces are rectangles.

A pyramid, however, has its base as any *n*-sided polygon where all the sides are triangles that meet at a common vertex, known as the apex. Imagine, quite literally, one of the pyramids in Giza, Egypt. Pyramids can have a triangular, square, pentagonal, or rectangular-shaped base. Prisms and pyramids are irregular polyhedrons that are essentially made up of polygons with different shapes so all of their components are not the same.

### Regular Polyhedrons

Platonic solids are regular polyhedrons, meaning all their faces, edges, and angles are congruent, regular polygons, and in which the same number of faces meet at each vertex. Platonic solids that we see in day-to-day life are dice. The five regular polyhedrons are: cube, tetrahedron, regular octahedron, regular dodecahedron, and regular icosahedron.

## Euler’s Formula

The relationship between the number of faces, edges, and vertices in a polyhedron can be presented in what we call “Euler’s Formula”. That formula is F + V – E = 2. You guessed it, F stands for number of faces, V for number of vertices, and E for number of edges. If you know two out of the three values, you can solve for the third quite easily. Let’s try it.

If we have a cube with 6 faces, 8 vertices, but don’t know the number of edges, we can plug in the information to solve for “E”.

6 + 8 – E = 2 becomes

14 – E = 2, and thus

E = 12.

Therefore, we know that there are 12 edges on that cube.

## At OMC

Polyhedrons are important to recognize not only to understand geometry better but also to help young math students understand and recognize objects in our daily life. At Online Math Center, we ensure that all students leave our school with a solid education of geometry concepts such as polyhedrons. Whether it comes naturally to a student or not, we offer classes and tutoring that can assist them in strengthening their foundation in mathematics. OMC works to ensure that every student reaches or surpasses their full potential in the world of mathematics.

Contact OMC today to help ensure your child gains a solid, complete understanding of the most important mathematical concepts, like polyhedrons, by signing them up for classes that are tailored to their specific educational needs.