When we say the area of a rectangle is the product of the length by the width is it a definition based on geometric intuition or is it a result? I know it is a result that we can find after defining integrals but before that was this a definition?
the definition of the area of a surface
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geometry
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1Generally, the **measure** (aka area) of a rectangle $[a,b]\times [c,d]$ is defined to be $|b-a|\cdot |d-c|$. – 2012-03-30
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0so areas of the rectangles as we know them are just **definitions** and not results that we can prove by geometric tools. – 2012-03-30
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1Yes, they are defined in that way. – 2012-03-30
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1If you consider only rectangles with integer-valued dimensions, you can certainly *define* the area of a rectangle to be the number of unit squares that fit in it, and then *prove* that this is equal to the product of the length and width. – 2012-03-30
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2Curiously, the usual modern approach defines the area of a rectangle *with sides parallel to the axes* like this, and derives this property for all other rectangles as a theorem. – 2012-03-30
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1You only need to define the area of the unit square to be 1. – 2012-03-30
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0yes and then the problem reduces to counting the number of unit squares inside the rectangle. what about the circle? – 2012-03-30
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0From the unit square you get rectangles (you need to use continuity for rectangles of arbitrary sizes), then triangles, then polygons, and finally circles by [exhaustion](http://en.wikipedia.org/wiki/Method_of_exhaustion) (you need continuity here again). – 2012-04-02