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I know that why we take area of a rectangle proportional to the product of its width and height. I also know why take area proportional to product. It is due to the theorem of Joint Variation. But it's really hard for me to make a visual concept or make sense of that how product of width and height give us the area of a rectangle?

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  • $\begingroup$ With regard to the area of the rectangle height into width formula is derived by simple integration $\endgroup$ – Heisenberg Dec 10 '14 at 10:06
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    $\begingroup$ That is completely circular, @Heisenberg. You defined the (Riemann) integral as the sum of the areas of smaller rectangles in the first place. $\endgroup$ – hjhjhj57 Dec 10 '14 at 10:24
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    $\begingroup$ Area is a very tricky business. More sophisticated approaches to area, e.g., Riemann and Lebesgue integrals, both require the axiom that the area of a rectangle is the product of its sides. You absolutely can't rely on any of these methods to prove the formula for area of a rectangle. $\endgroup$ – Ittay Weiss Dec 10 '14 at 10:45
  • $\begingroup$ Possibly related. $\endgroup$ – Lucian Dec 10 '14 at 18:46
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The area of a rectangle of sides $a$ and $b$ is defined to be $ab$. The motivation for this definition is coming from counting points in discrete rectangles. If you arrange $b$ rows of marbles, each row having $a$ marbles, then the total number of marbles you have in the rectangle is $ab$.

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