If I have a circle with an area of 100 units^2, and I divide it by $\pi$, how can I imagine that visually in my mind?
Since 100 / $\pi$ =~ 31.83, and the square of that is =~ 5.64, I currently visualize the result of 100 / $\pi$ being a box that is 5.64 x 5.64. And that for whatever reason, 3.14x of those boxes = a circle with a 100 units^2 area.
What is the take away? That a circle with a certain area can be broken down into 3.14 square boxes of equivalent area? How can I picture in my mind what that conversion even looks like?
These relationships seem arbitrary and un-useful. Is there a more meaningful relationship of these two facts?
Also, since this is kind of an open-ended question, is there a better place to ask such questions?
Let me first answer a slightly different question: how do you visualize that the area of a circle of radius 1 equals $\pi$? The answer I'll give was discovered by the ancients.
Before starting, we have to address what $\pi$ is, and I will take it to equal half the circumference of the radius 1 circle. In other words, I will define $$\pi = \text{(circumference)} \, / \, 2 \cdot \text{(radius)} $$
Imagine cutting the circle up into $1000000$ sectors, using $1000000$ equally spaced radii. Each of those sectors is very, very closely approximated by an isosceles triangle of angle $2 \pi / 1000000$ and of height 1. Number these sectors from $1$ to $1000000$. Lay out sector #1 in the coordinate plane with its base on the $x$-axis (the line $y=0$) and its opposite vertex on the line $y=1$, so it points "upward". Lay out sector #2 with its base on $y=1$ and its opposite vertex on $y=0$, so it points "downward", and so that it shares a side with sector #1. Continuing laying them out, alternating between upward and downward with each one sharing a side with the previous one, until all one million of them are layed out. The area that they fill out is very, very closely approximated by a rectangle whose height equals $1$ and whose base equals half of the circumference which is $\pi$. Therefore the area is very, very close to $\pi$.
So to answer your question directly, repeat this process except starting with a circle $C$ of area $100$. You will end up with a rectangle $R$ such that $$\text{Area of $R$} = \text{Area of $C$} $$ $$\text{height of $R$} = \text{radius of $C$} $$ $$ \text{width of $R$} = \text{half the circumference of $C$} = \pi \cdot \text{radius of $C$} $$ So, $$\frac{\text{Area}(C)}{\pi} = \frac{\text{Area}(R)}{\pi} = \frac{\text{(height of $R$)} \cdot \text{(width of $R$)}}{\pi} = \frac{\pi(\text{radius of $C$})^2}{\pi} = \text{(radius of $C$)}^2 $$