# 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...?

• In my opinion, "what would you get if you pretend $\pi=3$" sounds more like a research program in abstract geometry than a good way to present $\pi$ to kids. It certainly had me scratching my head and staring blankly into space for a few minutes. Just tell them pi is "3 and a little bit" while surreptitiously leading them to overshoot if you have to, but trying to work with "simplified circles" will just confuse everyone. Your creative exercices sound fun though. Note that in your example with the big sheet of paper you're saying the radius will have value 3, which is nothing to do with pi. – Joshua Pepper Mar 6 '14 at 22:03
• I, too, found the question in the first paragraph confusing. The is no precise answer: $\pi\neq 3$, so by the rules of logic, the statement "if $\pi=3$ then X" is true no matter what statement you plug in for X. I suppose the most reasonable interpretation is, "What figure has a circumference that is exactly 3 times its diameter?" Here again there is an infinite number of correct answers, but a regular hexagon is one of them. – Michael Weiss Mar 6 '14 at 22:26
• Blacksmiths used to use $\pi = 4$ for putting a metal rim around a wheel. That way, they would always have a bit left over -- better than not enough! I guess they could have used 3.2 or 3.5, but maybe whole number multiples were easier to work with under primitive conditions. – Dan Christensen Mar 7 '14 at 18:53
• If you want to teach them about the irrationality of $\pi$, I guess you could keep adding more points to the polygon and measure the ratio for each polygon, to get more precise approximations to $\pi$. – Vibhav Pant Mar 7 '14 at 19:08
• perhaps would be better suited for mathematics educators? – Liam Mar 2 '16 at 1:02

For helping kids (and adults!) understand $\pi$, I recommend the rubber chicken technique I describe in this answer.

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.

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 :)

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.

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

• I know this is quite old, but the answer you link to actually contradicts your last point: there is no Pi = 3, as Pi = 3.14... is the minimum value of Pi for any value of p. – Ron Feb 21 '16 at 3:25
• Yes that's true -- I think I had a factor of 2 floating around errantly in my head when I did this >.< – Emily Mar 2 '16 at 0:57
• Also, who says that we're defining 'the ratio of a circle's circumference to its diameter'? Maybe we're defining the value of the Wallis product. – Steven Stadnicki Mar 2 '16 at 1:04
• @StevenStadnicki It should be taken to be in the context of the question. – Emily Mar 2 '16 at 18:37

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.