Not $\pi$ - What if I used $3$? Teaching $\pi$ discovery to K-6th grade 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...?
 A: For helping kids (and adults!) understand $\pi$, I recommend the rubber chicken technique I describe in this answer.
A: As you know, PI is just a ratio:  perimeter of circle to diameter.
So, if you round PI down to 3, I would say you are still roughly
close to a circle. 
There is a cool book by Peter Beckman, A History of PI, which
also has a lot of neat references.
A: When we define $\pi$, we are defining the ratio of a circle's circumference to its diameter. But what is a circle? A circle is a set of points that are all the same distance from the origin. But what is distance?
In our everyday world, our notion of distance is interpreted in the Euclidean sense; that is, the distance between two points is the square root of the sum of the squares of the differences between the components, or, in two-dimensions:
$$d(x,y) = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}.$$
When we use this notion of distance, then we get a familiar looking circle. If we set the radius to be 0.5, then the circumference is $\pi$.
But... what if we defined distance differently? What if we defined distance as
$$d(x,y) = \max(|x_1-y_1|,|x_2-y_2|)?$$
Then, defining a circle around the origin, the set of all points that are 0.5 units from the origin looks like a square!
This square has a circumference of $4$ and a radius of $1$, so for this definition of distance, $\pi = 4$.
In fact, we can generalize this process. And if we pick the right notion of distance (called a norm), we can indeed find one wherein $\pi$ takes some other value between $3.14159...$ and $4$ exactly.
For more details, see the answer here: https://math.stackexchange.com/a/264312/31475
A: This is an interesting question, but you will not get anything other than a circle!  By drawing only 6-10 points you are approximating a circle, but that approximation only had to do with the number of points you used, not your estimate of $\pi$.  The only difference you will see is in results to the formulas that you mention.  One way that this could manifest is if you give students a value of circumference, and have different groups draw circles based on calculations with different values of $pi$.  You should see different sized circles, but they will all still be circles :)
A: Why not role various rubber wheels -- bicycles? -- of various diameters along the floor? Mark a point on each wheel so that you know when it has rolled exactly one revolution. Make sure the wheel turns freely or it could slip and throw off your measurements.
Simpler still, but maybe less exciting, use a flexible measuring tape to measure the circumference and diameter of various disks. Be sure to mark the center point to get accurate measures of the diameter.
A: Pi is the ratio between the circumference and the diameter. If you multiply a length (diameter) by pi you get the circle that has that length has a diameter. If you multiply by 3, you don't get the whole circumference, because you will be missing a piece. In order to get a circle again, you need to "curve" the space. Depending on whether you curve the space positively or negatively, the value of pi will become less that 3.14 or more. 
Maybe not what you were asking, but it is a fun way to teach kids about non-flat geometry.
