So, in ancient Mesopotamia they knew that they didn't really have the correct number ($\pi$) to determine attributes of a circle. They rounded to $3$. If you acted as though $\pi=3$, what shape would you get in our typical $C = 2 \pi r$ , $A = \pi r^2$ ? Would it be a polygon? A swirl? A sort of tear drop if you attempted to connect the two lines from where you started the circle and are ending it?
Related idea on which I would appreciate your thoughts: I am working on an activity to help kids "discover" $\pi$. This is for a homeschool group, so the kids range from 5-12 so I am making multiple levels to the activity.
One idea I had was to set up a big sheet of paper with a point in the center and having kids measure out 3 feet from that point.
Experiment 1: Have the group only create 6-10 points. We then connect the dots and essentially get a hexagon-decagon that might be slightly irregular looking.
Experiment 2: Have the group create as many points as they can. When we connect the dots, we get a super polygon - hopefully one with at least 60 points, and it will look even more like a circle.
Experiment 3: Hook a string to the central point, tie a pencil to the end, and have someone walk/draw the pencil in a full circle to get an authentic circle.
Discuss how this information might be applied. What if I wanted to make an enormous circular building? Additional discussion points...?