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In the definition of perimeter of a shape, we have a very clear view. Perimeter is nothing but the total length of the boundary of a given shape. So formulae for various shapes are also very clear.

But in case of area of a shape, we just have the idea that it tells about the size of the shape or more clearly, area informs us the amount of place which is covered by the shape in 2D surface. But how formulae for areas were derived as we know today for various shapes. For example, the area of a rectangle is length multiplied by its breadth (length×breadth). But who got the idea that the amount of space that a rectangle accommodates will be proportional to this mathematical value? I'm specially talking about this formula, because if we accept it as the basic formula then area formula for most other common shapes can be easily understood. Could different formula not be accepted, if the resultant mathematical values would have been proportional the sizes of the geometrical shapes?

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  • $\begingroup$ My guess is that long ago, people realized that if you arranged stones in $m$ rows of $n$ stones each, you use $mn$ stones, so the area of a rectangle that is $m$ stones high and $n$ stones wide is $mn$ stones. Today, we might not use stones as both units of area and of length, but it seems reasonable to think that was done long ago. $\endgroup$
    – Steve Kass
    Commented Feb 26, 2016 at 5:40

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One might think of the rectangle definition as being that "the area of an object is equal to the number of unit squares that can fit into that object" (obviously with some rescalings and a bit of nuance for curved objects - basically building area like a Riemann sum). If we were to reframe it as "the number of unit circles that can fit into that object" or "the number of equilateral triangles of unit side length that can fit into that object", then this would just be a scalar multiple of the square definition of area, and thus equivalent in most meaningful ways.

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  • $\begingroup$ Hmm.. perhaps, that's the idea. Area is the number of unit space enclosed by the region. $\endgroup$ Commented Feb 26, 2016 at 6:19

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