5
$\begingroup$

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?

  • 1
    Generally, the **measure** (aka area) of a rectangle $[a,b]\times [c,d]$ is defined to be $|b-a|\cdot |d-c|$.2012-03-30
  • 0
    so areas of the rectangles as we know them are just **definitions** and not results that we can prove by geometric tools.2012-03-30
  • 1
    Yes, they are defined in that way.2012-03-30
  • 1
    If 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
  • 2
    Curiously, 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
  • 1
    You only need to define the area of the unit square to be 1.2012-03-30
  • 0
    yes and then the problem reduces to counting the number of unit squares inside the rectangle. what about the circle?2012-03-30
  • 0
    From 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

2 Answers 2