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area
The area of a twodimensional figure is the amount of space contained within the figure. Area is typically measured in square units; i.e., if the area of a figure is 5 $\text{in}^{2}$, this means that, if five 1 inch by 1 inch squares are cut appropriately, they can be arranged so that they exactly cover the space contained in the figure without any overlapping. In formulas, area is almost always denoted using the letter $A$.
All examples provided within this entry are in Euclidean geometry.
For certain figures, area is quite commonly found by multiplying the lengths of two line segments which are related to the figure as well as perpendicular to each other. Below are some examples:
Also, in a regular $n$gon, each apothem is perpendicular to a side of the polygon. Thus, the formula $\displaystyle A=\frac{1}{2}aP$, where $a$ is the length of its apothem and $P$ is its perimeter, may be considered as another example.
Any threedimensional figure has a surface which is twodimensional. For certain figures, such as cubes and cylinders, this fact can easily be verified by cutting the surface and forcing it to lie flat. The surface area of a threedimensional figure is the area of its surface.
One method of determining the surface area of any threedimensional figure is by investigating how much paint would be required to cover its surface with exactly one coat of paint. (This works best if the paint is considered to have no thickness.)
There is another method of determining the surface area of a threedimensional figure. It works best on figures whose surfaces can easily be cut and forced to lie flat. Once this is done, the surface area can be obtained by determining the area of the resulting twodimensional figure.
For example, a cube is made up of six congruent squares. If each square has a side of length $s$, then the surface area of the cube is $6s^{2}$.
As another example, for a cylinder with radius $r$ and height $h$, its top and bottom, which are circles, can be cut off, and the remaining portion can be unrolled as a rectangle. The radius of each circle is $r$, so they each have area of $\pi r^{2}$. The rectangle has a width that is equal to the circumference of the circular faces, and its height is $h$. Thus, the area of the rectangle is $2\pi rh$. Therefore, the surface area of the cylinder is $2\pi r^{2}+2\pi rh$.
For some threedimensional figures, determining its surface area in this manner may not be very straightforward. For example, to determine the surface area of a sphere, one could try peeling an orange and making the portions of orange peel lie flat, but it would be very difficult to come up with the correct formula of $4\pi r^{2}$ from this procedure. The method of painting as described earlier works much better for spheres.
Remarks

When the shape of the geometric figure is complicated, the area can be computed using techniques from calculus. The idea is to break up the geometric figure into tiny squares. The area of the figure may be approximated by the total area occupied by these squares. The interesting thing is whether it is possible to get an exact answer if the squares are tiny enough. For all of the examples given above, using the tiny squares will give the exact answer.

The concept of area is a special case of a general concept called measure, or more appropriately, product measure.
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elementary definition of area
Due to a response that I sent Mathprof regarding a correction that he filed, I decided to add this entry. This is my humble attempt to try to define/describe area from an elementary (precalculus) point of view. I realize that some people may want to add to or change this entry, so I will leave that option open for a while. Since I want this entry to be from an elementary point of view though, I would greatly appreciate it if derivations from calculus (or beyond) are left out of this entry. On the other hand, links to entries that deal with concepts from calculus or beyond are welcome, so long as they relate to the entry. For example, I used the word "cover", which links to the topology definition, but I feel that this link is appropriate.
I would highly encourage people to add entries that show, at least in a special case, that the elementary concept of area corresponds to the concept of area as presented in calculus. I may file this as a request at some point.
Re: elementary definition of area
I think what you're doing is a good idea. It'd be nice to see the concept explained from different points of view as well as at different depth level. I would suggest that at these elementary levels it would be a good idea to include some pictures, especially when they are geometrical in nature.
Pictures!!! (was elementary definition of area)
>I would suggest that at these elementary levels it would be a good idea to include some pictures, especially when they are geometrical in nature.
I thoroughly agree! Unfortunately, I remain absolutely clueless as to how to include pictures on PM. Specifically, I do not know how to create pictures in TeX. I asked about this a couple of times here on PM and received responses that were way over my head. Even more frustrating is that I have adopted some entries in which former owners included pictures, but I cannot retrieve the code from them! Any help that I can get on this (most likely including examples of code that produces certain pictures) would be very much appreciated.
Warren
PSIf anyone decides to add pictures to any of my worldeditable entries, ***please*** let me know how you did it.
Re: Pictures!!! (was elementary definition of area)
From a cursory glance of PM entries, it seems that the ``pstricks'' package is quite popular and might be easier for those who can't run LaTeX standalone on their computer. But I don't use this package, so I can't really help you with it.
Recently I have been doing my diagrams using the Python programming language. IMO this is the most powerful technique, but like most of math and computer science, powerful things have a steeper learning curve than simpler things :)
Before that I have used MetaPost, and if you willing to dig into that,
I have some examples available at, which are used in PM entries:
http://svn.goldsaucer.org/repos/PlanetMath/Interval/intervals.mp
http://svn.goldsaucer.org/repos/PlanetMath/TriangleSolving/triangle.mp
http://svn.goldsaucer.org/repos/PlanetMath/PerpendicularBisector/constr...
http://svn.goldsaucer.org/repos/PlanetMath/QuadraticInequality/parabola.mp
http://svn.goldsaucer.org/repos/PlanetMath/LeibnizEstimateForAlternatin...
Note some of these may be somewhat complicated; you might want to follow along with one of the many MetaPost tutorials available
(this one is my favorite: http://remote.science.uva.nl/~heck/Courses/mptut.pdf)
or feel free to ask me for explanations about specific sections of the code.
I may add some diagrams to your entry if nobody does them (and when I get time to do so).
// Steve
Re: Pictures!!! (was elementary definition of area)
I have added pictures for the triangle, parallelogram, and the ellipse. (Technically, CWoo added the picture for the triangle.) I would like pictures to show a cube being cut up into six squares and a cylinder being cut up into two circles and a rectangle. I will try to figure out if/how I can do this in pstricks, but if anyone else comes up with ideas before me, please feel free to add them. Especially entertaining would be pictures of someone peeling an orange and trying to flatten the orange peels. :)
Re: Pictures!!! (was elementary definition of area)
Pstricks isn't really latex which makes it awkward to use because you right in postscript but using latex notation, I kind of hate it.
My goto methods are:
xypic  stinks really although the xymatrix method is reasonable But now I'm used to it.
(see for instance any of the entries on universal mapping properties, such as free algebra etc.)
On linux/Unix/Apple:
GNU plot  very very easy, and exports the proper LaTeX commands.
(see for instance "Enumerating groups" which has such a graph.)
On Windows:
TclTk Draw, antique but makes nice encapsolated postscript diagrams vectorially rathar
than pixels.
I don't use Windows any longer so I don't have examples of this to share.
minimalist behavior (was elementary definition of area)
After adding pictures to some of my entries, I have noticed the following pattern: In html mode, there is a tendency to display as little of the pspictures as possible so that all of the text is displayed. Is there any way to prevent this minimalist behavior from occurring in html mode?
Peeling oranges
> Especially
> entertaining would be pictures of someone peeling an orange
> and trying to flatten the orange peels. :)
Quite a lot of work, but may be next time I peel an orange, at least I take a picture :)
However, I think the present definition of surface area is not so good, because as we all know, it is impossible to flatten any portion of a sphere, and practically speaking, I don't think it is possible to get any good accurate estimate of the area that way. A more workable definition, I think, is to consider the amount of paint that would be used if one were to paint an orange. If the thickness of the paint layer is 0.5mm, say, then the surface area is the amount (volume) of paint used / 0.5mm.
// Steve
Re: Painting oranges blue
I definitely agree that the "definition" for surface area that I provide is not the best, but the method of taking figures apart and flattening them to determine their surface area is a good one for students to know in many situations. Therefore, I would like the examples for cubes and cylinders to remain pretty much as they are, but I may reorganize the entry so that surface area is better defined.
Could we add pictures of someone painting an orange blue?
Re: Peeling oranges
And the thinner the layer of paint, the more accurate of an estimation of that area is.
Eureka! (was minimalist behavior)
I finally found a way to make the graphics display correctly in html mode: I added ``labels'' that are barely perceptible. (In page images mode, I cannot see them at all.) I am so happy that this finally works!
There has been discussion about the ``definition'' of surface area as it currently stands in this entry. Once that is fixed to my satisfaction, I will take away the world editable option. In other words, last call to make changes to this entry!
Good night everybody.
Warren
Re: Eureka! (was minimalist behavior)
> There has been discussion about the ``definition'' of
> surface area as it currently stands in this entry. Once
> that is fixed to my satisfaction, I will take away the world
> editable option. In other words, last call to make changes
> to this entry!
The issue with your definition is that it only works for surfaces
which are isometric to a plane. For instance, cylinders and cones
are isometric to planes, so we can get their surface area by laying
them out upon a plane but spheres are not, so we do not have recourse
to this approach to define the area of a sphere. Instead, to define
the surface area of a sphere we must do things like approximate it
by a poyhedron or take the volume of a thickening of the sphere,
divide by the amount we thickened the sphere and take the limit as
this quantity goes to zero.
The reason I didn't edit the entry directly is because I am not sure
whether talking about all this fits within the scope of the entry as
you planned it as an elementary introduction to the subject. If you
think it is appropriate, include it or let me know so I can add it.
If not, maybe just say a few words about this not working for all
surfaces and refer the interested reader elsewhere for a discussion
of what goes wrong with spheres,.
Re: Eureka! (was minimalist behavior)
Raymond,
You may want to take a look at this string of posts:
http://planetmath.org/?op=getmsg&id=14751
Warren
Re: Peeling oranges and painting oranges blue
I have adjusted the entry ``area'' in order to take this painting strategy into account. I am leaving it world editable for a while in case there are ideas that I have not explained clearly.
Pictures of people peeling oranges and/or painting oranges blue are still welcome! :)