Bases and Faces

Date: 12/05/2001 at 14:07:24
From: Madison
Subject: Bases and faces

In math class we are learning about polygons and I can't figure out 
the difference between a base and a face on the shapes we are 
learning. My teacher says a face is a base but a base is not a face. 
What is the difference? How many bases does a cube have, and how many 
faces does it have?

Date: 12/05/2001 at 14:48:31
From: Doctor Peterson
Subject: Re: Bases and faces

Hi, Madison.

We generally use these terms in different settings.

A cube on its own has six faces. Here we're not picturing it set on a 
table, but just sort of floating in space, so that all six faces are 
equal, and we don't think of any of them as special.

When we are talking about how to calculate the area or volume, we 
usually think of one face as the "bottom," and call it the base, as if 
we were setting it down on a table to measure it. The "top" may be 
seen as "the other base," since they are identical, and the other 
faces are the "sides." So when you set the cube down, it has one base 
(or two if you prefer) and four sides.

It really doesn't make any difference which face you call the base 
when you talk about a cube, because they are all the same. But for, 
say, a box (a rectangular parallelepiped), you have three different 
lengths, and by choosing a base you are deciding which two lengths to 
use to find the area of the base, and which length to call the height. 
You can choose any face to be the base, and you will get the same 
answers.

For some shapes, such as a cone, there is only one flat surface to use 
as the base. Then no questions arise (until you ask whether the curved 
"side" can be called a face - but that's another question, and we've 
answered that in our archives).

Incidentally, I am assuming you are learning about polyhedra, not 
polygons; the latter are flat. A cube is a polyhedron.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/