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I want to rigorously prove the idea that Base*Height=Area works (I do realise there are shapes which do not satisfy this equation). I think I can see why it works for integer values, but I want something that gives me the correct intuition for all positive numbers.

Because of my level, I would appreciate if nothing beyond calculus is used to prove this. (Although I doubt calculus is needed).

Edit: As asked, my level is multivariable calculus, or CALC III in the US I guess. I want to prove this basically for a rectangle.

Thanks a lot.

DLV
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  • That formula is only true of specific shapes as circles, triangles and a few other shapes would not work with that formula you do realize, right? – JB King Jul 29 '14 at 00:06
  • Yes, do you think I should include this information in my question? – DLV Jul 29 '14 at 00:07
  • Yes as there is something to be said for which shapes are you wanting to consider: Rectangles, parallelograms, squares? – JB King Jul 29 '14 at 00:09
  • "Because of my level, ...". It would help if you explained exactly what your level is. – Blue Jul 29 '14 at 00:09
  • What about the area under the function f(x) = 1 if x is irrational, 0 if x is rational? – djechlin Jul 29 '14 at 00:12
  • @Blue I've edited my question now. – DLV Jul 29 '14 at 00:12
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    The subject you'd want to research is measure theory; however, that won't help you much as the areas of rectangles are actually defined as the product of the lengths of its sides. –  Jul 29 '14 at 00:17
  • @Bryan But I mean, before the definition was out there, how did they know this was true? – DLV Jul 29 '14 at 00:38
  • http://math.stackexchange.com/questions/653264/proving-the-area-of-a-square-and-the-required-axioms?rq=1 links to a page that shows how this problem reduces to knowing that the area of a square of side $s$ is $s^2$, and also has a reference to a book that is said to prove that fact. – David K Jul 29 '14 at 00:47
  • @David fair question but now you're asking a history of math question, an anthropological question, a philosophy of math question, etc. it's so basic it's an axiom; so you certainly won't be able to prove it with anything more intuitive-seeming. – djechlin Jul 29 '14 at 00:47
  • @djechlin I can see how it was axiomatic for integer numbers, but it simply baffles me when I consider irrational numbers. I wonder how they did it. I guess I'll save this question for later on. – DLV Jul 29 '14 at 01:54
  • @David other axioms are inequality-related (taller rectangles of the same width should have larger area), and summability properties (which gives you rationals, e.g. using 1/2+1/2=1). But yeah it's complicated; see measure-theory or at the least look up a riemann integral. – djechlin Jul 29 '14 at 02:28

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