How can I visually imagine the area of a circle divided by $\pi$? 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?
 A: 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,
$$\frac{\text{Area}(C)}{\pi} = \frac{\text{Area}(R)}{\pi} = \frac{\text{(height of $R$)} \cdot \text{(width of $R$)}}{\pi} = \frac{\pi(\text{radius of $C$})^2}{\pi} = \text{(radius of $C$)}^2
$$
A: Assume your circle is made of pancake dough, clay or any such thin stretchable material. Take the top point of circle and slightly pinch it in and pull out the points at diagonals until it reaches out to the corner of square. You added some area and removed the same area, you only changed its contour but not its total area...
circle to square
