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area

Defines: 
surface area
Type of Math Object: 
Definition
Major Section: 
Reference
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Mathematics Subject Classification

51M25 no label found51-00 no label found

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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.

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.

>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

PS--If anyone decides to add pictures to any of my world-editable entries, ***please*** let me know how you did it.

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 stand-alone 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.gold-saucer.org/repos/PlanetMath/Interval/intervals.mp
http://svn.gold-saucer.org/repos/PlanetMath/TriangleSolving/triangle.mp
http://svn.gold-saucer.org/repos/PlanetMath/PerpendicularBisector/constr...
http://svn.gold-saucer.org/repos/PlanetMath/QuadraticInequality/parabola.mp
http://svn.gold-saucer.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

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. :-)

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 go-to methods are:

xy-pic -- 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:
Tcl-Tk 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.

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?

> 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

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?

And the thinner the layer of paint, the more accurate of an estimation of that area is.

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

> 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,.

Raymond,

You may want to take a look at this string of posts:

http://planetmath.org/?op=getmsg&id=14751

Warren

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! :-)

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