# Is it possible to employ the circle as a primitive concept to measure the area of the rectangle?

Historically, it seems that we have found more practical uses for the area of the rectangle than the area of the circle. The definition of the area of the rectangle was given and used as a primitive concept, with the area of the rectangle one can easily find the area of the triangle with a method such as the one given in this question, thus making the triangle the second geometrical object in the hierarchy (I guess that this is true in the context of euclidean geometry).

Having this area defined, it seems that we tried to find a definition for the area of the circle that was dependent on the definition for the area of the rectangle and triangle, then it was found the method of exhaustion, which employs regular polygons (which can be decomposed into triangles), then the given method is employed to find an approximation of the area of the circle with the sum of the area of these triangles.

So, given the hierarchy:

$$\begin{matrix} {1}&{Rectangle}\\ {2}&{Triangle}\\ {3}&{Circle} \end{matrix}$$

Has someone tried to find the area of the rectangle and triangle by using the area of the circle as a primitive? Is there a definition of the circle that is not dependent on the rectangle and triangle and that could be used to find the areas of them (triangle and rectangle)? Perhaps it's the dumbest thing to do on a lifetime, but I'm curious if such a thing exist and/or it has revealed something of value.

• So are you thinking like, if one unit area is defined as the area of a unit circle, what would happen? – peterwhy Dec 1 '13 at 8:39
• Yes. I guess this is it. – Billy Rubina Dec 1 '13 at 9:31

On one interpretation, this is the idea behind Hausdorff measure, which constructs a notion of "volume" given only a notion of "distance". The circle is the unit ball of the usual Euclidean notion of distance, so in a sense this is taking the circle as fundamental. In $\mathbb R^n$, Hausdorff measure is the same as the familiar notion of volume (up to a scaling factor); the real benefit of the construction is that you can apply it in other metric spaces, where you might not have any preconceived idea of volume.
If we would have first defined the primitive as a circle , what would have happened is that we would have divided the rectangle into small triangles to fit it inside the circle . Either way you do it $\pi$ will come into the picture .