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I'm starting a new personal project and I need a little bit of helping on setting it up.

First off I'm trying to determine the area of a rectangle that is 1 mile in width and the circumference of the earth minus a mile in length(like a great circle minus a mile). I understand I can solve this with an equation but I want to map it.

Anyways I don't know what projection I need for this and how to get started.

Is there a general earth projection in ArcMap? Do I even need a projection if it is the whole earth? Do I need to worry about equal area distortion? and etc.

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    To a high degree of accuracy, the formula area = length * width will work beautifully. Assuming a circumference of 24,901 miles gives 24,900 square miles as the area. Now you can focus on the mapping. Although you don't need to project the whole earth--that's impossible to do in a continuous fashion, anyway--you do need to project the region around the rectangle itself. And you're on the right track with equal area projections if you would like to visually compare your rectangle's area with areas of other regions on the same map. That should help narrow your options.
    – whuber
    Commented Apr 18, 2014 at 19:47
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    I'm having a very difficult picturing this. Do you mean a 1 mile wide strip that encircles the earth (aka crosses the poles), but excludes a 1 mile "north-south" section? On a sphere or an ellipsoid?
    – mkennedy
    Commented Apr 18, 2014 at 19:54
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    Projection: Mercator. In particular one where the line of tangency goes down the center-line of the long axis of your polygon.
    – user23715
    Commented Apr 18, 2014 at 20:31

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Every time you want to display information from the surface of the Earth on a map (or on a flat screen), you are doing some kind of projection. It is however not necessary to use a projection for analysing your geographic data.

When you work in a projected coordinate system, different types of distortions may occur, and it is impossible to build a projection that preserves both the shape (conformal projection) and the area (equal area projection) of the surface being mapped. There is thus no such thing as "general projection" of the Earth that would make everyone happy. Some are compromise (Robinson, Van de Grinten...) but none is perfect.

So if you want to compare the area, equal area projection are the best choices (Sinusoidal or cylindrical area can be used for global maps), but then a rectangle on the ground may not look like a rectangle on the map anymore. Mercatoe, on the other hand, is a conformal projection.

Finally, you should also consider that your polygon will probably look like a line on your map when you are at full extent. And if you go accross the pole, the projection that I mentioned will be discontinued.

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