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I need an explanation of the difference between parameter and variable in the following example. In extremal geometric problems when we want to find the object having some extremal property, say maximal area, we choose an angle or side and express the area in terms of it. Is the angle (or the side) parameter or variable?

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I would be inclined to say that a parameter is a specific type of variable; in your situation, since you have other quantities expressed in terms of it, I'd call the "it" a parameter, but since you have expressions with some label for that unknown quantity, "it" is still a variable.

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