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The definition of area usually include the area of a rectangle definition. Can one replace it with "the area of a square of side $a$ is $a^2$"? That is can one find the area of a rectangle in this case?

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  • $\begingroup$ See my answer to a similar question. Basically, you start with the axiom that the area of a $1\times 1$ square is $1$, and use further axioms to extend the definition of area to arbitrary rectangles. $\endgroup$ – TonyK Sep 2 '14 at 14:52
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I've seen something like this in a textbook. Take a square and add $2$ lines to form smaller squares in opposite corners and 2 congruent rectangles. If the length of a side of one of the inner squares is $a$ and one of the other inner square is $b$, the outer square has sides of length $a+b$. So if you take the area of the outer square, subtract the areas of the other squares, and divide by $2$, you should get the area of one of the rectangles with sides of length $a$ and $b$.

$$\frac{(a+b)^2-a^2-b^2}2=\frac{2ab}2=ab$$

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