Polyhedra

What is a polyhedron? Polyhedra are solids whose faces are polygons. So the faces could be anything from squares to the “L” shaped figures in Tetris. As long as the 3-D shape has all straight edges and flat faces, its a polyhedron (Boswell, Lee, and Larson)

This is a cube. Cubes are polyhedrons.

Of course, there are some shapes that are not polyhedra. These are any shapes that have a curved or any not-flat face. Some examples are cylinders and cones. Their circular base is not a polygon, and its lateral surface isn’t flat (Boswell, Lee, and Larson).

This is a cylinder. Cylinders are not polyhedrons.

All polyhedra have three features that define the shape; faces, vertices, and edges. Faces are the shapes that make up the shape itself. For example, a face of a cube is a square. Edges are the line segments made when two faces meet. Last, a vertex is a point where at least three edges meet. In some circumstances you might need to find out the number of each of these features a certain polyhedron has. You could just count them manually, but that’s just a pain. There is a theorem that can make it a lot easier. It is called Euler’s Theorem, and it makes it really simple to figure out the number of faces, edges or vertices (Boswell, Lee, and Larson).

Euler’s Theorem: F + V= E + 2

F= faces

E= edges

V= vertices

Basically, the sum of the number of faces and the number of vertices is equal to the number of edges added with two.

Here are some examples of how to use this theorem.

You have a figure with 90 edges and  32 faces. Find the number of vertices.

F + V = E + 2 Write the equation

(32) + V = (90) + 2 Substitute in what you know

32 + V = 92 Combine like terms

-32 -32 Subtract 32 from both sides of the

equals sign

V= 92 + (-32) After subtracting from both sides

V= 60 Combine like terms

There are 60 vertices in this shape.

See, you don’t need to see the shape to figure out the number of faces, edges, and vertices.

There are other terms you can use to categorize polyhedra.

One is regular or irregular. Regular polyhedra are polyhedra whose faces are all the same and congruent. Unlike this, a non regular polyhedron’s faces are not all the same (Boswell, Lee, and Larson).

This is an icosahedron. They are regular.

This polyhedron is not regular.

The next is convex or concave. Convex shapes are shapes that look full and complete. “A polyhedron is convex if any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron. (Boswell, Lee, and Larson)” Concave 3-D shapes kind of look like there was a chunk punched out of it, and cave in.

This shape is concave.