# Face

A flat surface of a solid figure; A polygon that is a flat surface of a three-dimensional figure.

Example:

A face is one of the polygonal surfaces of a polyhedron.

FAQs

- Are the circles at the top and bottom of a cylinder called faces?
- Why do we need to call the sides of a 3-D shape "faces"? What's wrong with just calling them "sides"?

## Meaning

A face is one of the polygonal surfaces of a polyhedron.

This rectangular prism has six faces. We can see three of them (blue, purple, and green); the other three are hidden from view. This polyhedron has eight faces. We can see five of them -- four rectangles (of which the blue, purple, and red ones appear to be square), and a green L-shaped six-sided figure (hexagon). The other three faces -- the bottom rectangle, the tall right-side rectangle, and the back L-shape -- can't be seen.

If we build out of cubes, we might see extra lines on the surface, like this: , but there are still only six faces, the green rectangular face, the purple rectangular face, the square blue face, and the three we can't see. The extra lines we might draw on these faces do not create extra faces, just like the extra point on this triangle does not mean it has four sides.

Not faces:By the definition given above, surfaces that are not polygons, even if they are flat, would not be called faces. So, by that definition, the circular surfaces of a circular cone or cylinder are not faces. But some texts and even state tests use a looser definition, using "face" for any flat surface of a solid, therefore including the two circular flat surfaces of the cylinder. The term "face" would never be applied to the "wrapped-around" surface of the cylinder.Think Math!uses the "stricter" definition, but which use you choose is not mathematically important issue.

**Under construction:** more to come...

## Mathematical background: surface, face, and net

We can think of the surface of a polyhedron as a boundary separating the inside (interior) of the solid from the outside (exterior), just as the sides of a square separate the inside from the outside.

Imagine the surface of this rectangular prism as a wrapper, and imagine unfolding it.

As you unfold, you might see this .

Looking down on the wrapper when it is entirely flat on the table, you would see this .

The resulting pattern of polygons is called the net of that rectangular prism. It has the same number of polygons as the solid had faces, one polygon for each face of the solid.

Similarly, this solid with eight faces has a net composed of eight polygons. Because of this perfect correspondence, the word *face* is often used to refer to the polygons in the net.

Elementary school students who have had some experience examining prisms of various kinds can figure out the number of faces a prism has *without counting*, just from knowing about the prism's base. The prism has two bases, all one needs to know is how many *other* faces it has. That is determined by the number of sides on the base. A hexagonal prism will have the two hexagonal bases, and six more faces, one for each side of the hexagonal base, for a total of eight faces. Children are often quite pleased when they discover, on their own, that they can figure out the number of faces without counting. In a similar way, they can figure out that the number of faces on any pyramid just by adding 1 (for the base) to the number of sides on the base. So, a square pyramid has five faces: the base itself (a square) and four triangular faces, one for each of the sides of the square base.

**Under construction:** comparing faces to vertices (prism, V greater than F; pyramid, V equals F; antiprism, V less than F)

## What's in a word?

Why do we need "face" when "side" seems to work just as well?

In two dimensions, terms like "side" and "corner" *are* clear enough, but in three dimensions, they become ambiguous. Casual English uses the word "side" in too many ways! We might refer to the two blue, two green, and two purple faces of as six "sides," but if we imagine a *room* of the same shape, we would credit it with only four "sides" because, in that context, we don't include the ceiling and floor. Moreover, each of the faces of the prism is a rectangle with four "sides," giving the word side yet another meaning -- the *line segments* at which the walls of the room (or the faces of the prism) meet. Too many possible meanings!!

So we use a new word to make communication clearer.

* Under construction: Etymolygical connections are just notes, for now. More coming... Will include relationship among *face

*,*surface

*,*superficial

*,*facet

*of a diamond (not the mathematical technical meaning of facet).*

## Related mathematical terms

See also edge, vertex, surface area