# The Ultimate Pi Day

**By Liz Hutchings**

*Faculty Member, Mathematics at American Public University*

As a college math instructor and a homeschooling mom, Pi Day is one of my favorite holidays. Whether it is a real holiday or not is up for debate, but in my home this year’s Pi Day will definitely be one to celebrate.

In case you haven’t heard, this Pi Day is special because it will not only be March 14 (3.14), but we can get as accurate as 3/14/15, 9:26 and 53 seconds = 3.141592653. That is just incredible and will never happen again in our lifetime.

**Defining Pi**

Just as a refresher, pi is that funny Greek symbol, Ï€, that you tried to forget about in high school math. You probably wondered why pi existed when you always just used 3.14. So, why can’t pi just equal 3.14? Why all this fuss over something that everyone tells you just to approximate?

In math, we like to come up with formulas for relationships. Finding areas and perimeters are a couple of examples of formulas we like to have on hand. When working on finding these formulas mathematicians had no trouble finding the perimeter of a rectangle because the relationship was simple: the perimeter was equal to the sum of the length of the sides or 2L + 2W.

When looking at the circumference or perimeter of a circle, it cannot be easily measured by hand because it is not a straight line. In order to accurately find the circumference of a circle, you need a relationship to a measurable straight line. That straight line is the diameter of the circle, which cuts it cleanly in half. There is a consistent relationship between the length of the diameter and the circumference of the circle.

As this relationship was studied, mathematicians struggled to find its exact value. Different examples you may have used include 3, 3.14, 22/7, but no matter which approximation is used, none are perfectly accurate. This is where pi comes into play.

Pi is the ratio between the diameter and circumference of a circle and is irrational, meaning is cannot be defined by decimal or fraction. The only accurate definition of pi is:

**History of Pi**

The ancient Babylonians and Egyptians studied pi because they needed to know the distance around circles for construction of buildings. Early mathematicians approximated pi as 3; this approximation is found in the Bible. It was simple and easy to use and not too far off.

The Egyptians found that using 3 was convenient but not precise and attempted to establish the ratio by relating it to the area of the square. This method yielded an approximation of 3.1605. Without computers or calculators that is just not too shabby.

Before 200 B.C., the Greeks used inscribed polygons to approximate pi. They found that the more sides the polygon had, the closer their approximation would be. Archimedes was able to narrow the field even further by finding that pi was between 3.1408 and 3.14285.

In modern times, mathematicians became a little obsessed with calculating pi and, in 1873, William Shanks published his calculation of pi to 707 decimal places. Only 527 of those decimal places were correct, but he spent years working on his approximation. To simplify and ensure accuracy we use the symbol Ï€ to represent that irrational value. This symbol was first used by William Jones in 1706.

**Fun Pi Activities**

To illustrate this point, and as a fun activity for *your* Pi Day celebration, take a really long rope, string or yarn and create a circle on the floor. You can make it as big or as small as you like. Then, take another piece of rope and cut it to the length of the diameter, the longest distance across the circle.

Next, we want to see how the length of the diameter compares to the circumference, the distance around the circle. Using your diameter rope as a guide, cut two more of the same length, so you have three diameter length ropes. Lay these ropes end-to-end around the original circle. They should all fit, and there should be a small gap after all the rope is used.

If you have kids, it is fun to give each of them one of the ropes to lay down. The ropes show us that the circumference is a little more than three times the diameter. This exercise is great for young and old to help understand where pi comes from, and it helps you see the problem that mathematicians faced when trying to define this relationship.

Another favorite activity of mine is to fill a cylindrical jar with a round object (marbles, M&Ms or something else round) and ask everyone to guess how many objects are in the jar. I am sure you have seen this activity before, but have you ever use pi to help you figure it out?

Remember that the volume of a cylinder is *Ï€r ^{2}h*, so you can measure the radius in the number of objects, count the height and multiply! Lift up the jar above your head and look at the bottom—it’s a circle!

- Count the number of M&Ms (or other objects) it takes to get from one side of the circle to another–that’s the diameter! Half the diameter is your radius,
*r*. - Put the jar back down and count how many M&Ms it takes to get, in a straight line, from the bottom of the cylinder to the top–that’s your height,
- Plug your radius and height into the formula,
*Ï€r*and you’ll have a smart approximation of the total number in the jar.^{2}h,

It won’t be perfect unless your objects are perfectly placed (not realistic), but it should get you pretty close, and after you announce the winner you can do the calculation for those who are old enough to understand it.

You should, of course, have pie or some other desert with a Ï€ in it. I think this year I will do cup cakes and put the approximation of pi one digit at a time on the top.

Here are some other fun ideas you may like to try:

- Find out where your birthday is in pi: http://www.facade.com/legacy/amiinpi

- Create a pi chain or see who can memorize the most digits of pi (in order).

- Read some fun pi books together. My favorite is ‘Sir Cumference and the Dragon of Pi.’

- Enjoy some pi poetry: http://www.eveandersson.com/pi/poetry

- Sing some pi day carols: http://teachpi.org/music/pi-day-carols

Or make up your own activity, but do not let this great holiday pass you by. Happy Pi Day to all!

**About the Author**

*Liz Hutchings loves math and always knew she wanted to be a math teacher. She completed both her BA in math education and MS in mathematics at Brigham Young University. She currently teaches online as a full time faculty member at APU.*