Approximate the Value of Π in Your Kitchen

A few introductory facts about π.

The number π is probably the most well known mathematical constant. It is defined as the ratio of a circle's circumference to its diameter and is commonly approximated as 3.14159. Hippocrates Of Chios was the first to prove that this ratio is constant for any given circle.

Being an irrational number, π cannot be expressed exactly as a fraction. Or equivalently, its decimal representation never ends and never settles into a permanent repeating pattern (as we understand so far).

It has been represented by the Greek letter π since the mid-18th century, though it is also sometimes spelled out as p-ai.

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes. This polygonal algorithm dominated for over 1000 years. That's the reason why π is sometimes referred to as Archimedes' constant.

Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle. He successively doubled the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223 / 71 < π < 22 / 7 (that is approximately 3.1408 < π < 3.1429). It is said that Archimedes used 22 / 7 as an approximation of π.

Step 1: Materials

In this instructable you will learn how to approximate Archimedes' Constant π in the comfort of your kitchen.

This is a fun, educational project to carry out with your kids, grandchildren, nieces or nephews.

We are going to use a small model and then we are going to scale it up.

You will need :

several Oreo type biscuits (or any other circular shaped ones)
an A4 sheet of paper
a pencil
a ruler
a calculator
cling foil (optional)
printer to print the circular template (download my template or draw your own)
a beam compass (for the bigger model)

For the construction of the beam compass look at my other instructable :

Extra Large Beam Compass.

Step 2: Download the Template for the Small Model

The biscuits I used measured approximately 4 cm in diameter (and were delicious!).

I created a simple template to put the cookies on.

The template was created using Xfig 3.2 . It is a GNU/Linux schematics program.

Download my template or create your own using paper and compass.

Step 3: Small Model - Measure the Circumference

Print the template and put it on the kitchen table.

Begin by putting the biscuits in circular order, following the guides.

Just for now, resist the temptation to eat them. Leave it for the end of the activity.

Look at the above pictures. In my case the last biscuit leaves a small space. Measure that distance with your ruler and write it down. My measurement was 0.7 cm.

Also measure how many whole biscuits you used. I used 12.

So the circumference of the dashed circle will be :

C = 4 x 12 + 0.7 = 48.7 cm.

Step 4: Small Model - Measure the Diameter

Now put some biscuits along the diameter line of the template.

The edge of the first biscuit must barely touch the dashed line.

In my case, I used 3 whole biscuits and a portion of the fourth.

The fourth biscuit was cut at 3.6 cm in order to fit.

So the diameter of the dashed circle will be :

d = 4 x 3 + 3.6 = 15.6 cm.

Step 5: Small Model - Data Crunching

To calculate π, divide C with d. In my case :

π = C / d =

= 48.7 / 15.6 =

= 3.12179...

Since the real value for π is 3.14159... we get :

(3.12179... - 3.14159...) x (100 %) / 3.14159... =

0.63025 %... or roughly 0.63 % deviation from the real value. That's a decent approximation!

Step 6: Scale Up the Model

Let's repeat the measurement, only this time we will use a bigger template and a lot more biscuits.

The bigger circle has diameter 60 cm and the smaller 56 cm. They were drawn using my big beam compass on a piece of cardboard. You can see how I made this device, viewing my other instructable :

Extra Large Beam Compass

Step 7: Big Model - Measure the Circumference

Before laying the biscuits I applied several pieces of cling film along the path because the cardboard was dirty.

As before, lay the biscuits along the circular path. I used 42 whole and 3.6 cm of the 43th.

So now the circumference will be :
C = 4 x 42 + 3.6 = 171.6 cm.

Step 8: Big Model - Measure the Diameter

Now put some biscuits along the diameter.

In my case, I used 13 whole biscuits and 2.7 cm of the 14th.
So the diameter will be :

d = 4 x 13 + 2.7 = 54.7 cm.

Step 9: Big Model - Data Crunching

To calculate π, again we divide C with d. In my case :
π = C / d =

= 171.6 / 54.7 =

= 3.13711...

Since the real value for π is 3.14159... we get :

(3.13711...- 3.14159...) x (100 %) / 3.14159... =

0.14260%...

or roughly 0.14 % deviation from the real value.

The more biscuits, the merrier!

Step 10: Conclusions

In this activity we used the polygon approximation method in order to approximate the value of π. This method converges slowly and requires polygons with a large number of sides (more biscuits and/or smaller biscuits in our example). There are other algorithms that converge quicker towards the answer such as infinite series, continued fractions, computer iterative algorithms, etc.

The deviation from the real value of π is justified, if we consider the fact that the cookies were not perfect cylinders and our crude measurements have finite precision.

But this can't lessen the fact that we measured the constant πin our kitchen with decent accuracy. Bravo!