Determining the Value of pi Through Experiment

Motivation: Pi () is an irrational number, meaning that we cannot write it in terms of a fraction, and its decimal form contains an infinite string of numbers in a seemingly-random order. Why is pi important and if its decimal representation goes on forever how can we really compute a value of pi? You have probably used in school to figure out the area and circumferences of circles. This is because is the ratio of the circumference to the diameter in a perfect circle.

Objective: Despite all the mystery, we can attempt to calculate the value of from data collected in an experiment and in the process gain insight into why is such a special number. Since there are no “perfect” circles to measure, we must measure sets of different circles many times in order to get a picture of what an average (hopefully closer to perfect) looks like. In this activity, we will design and perform an experiment to collect data to be used in the calculation of

Materials:

  • Set of variously sized balls, ranging from ping-pong balls, to basketballs, to hula-hoops.
  • Meter-stick
  • Length of string or cord
  • Pencils
  • Sheets of paper
  • Other types of measuring equipment you request (within reason). You are designing an experiment. If you don’t think you have the right tools for the job, just ask for what you need!

Instructions:

1.)Break into your small groups and do all the following steps together.

2.)Determine methods for measuring:

  • Circumference of the objects
  • Diameter of the objects

3.)Record those methods you developed in the following section marked “Procedure”.

4.)Create a list of all the materials you have used in your procedure.

5.)Create a table for your data containing all of what you think is relevant information that you have collected

6.)Complete the Discussion Questions that follow

Procedure:

Data table:

Discussion Questions

In your measurements, did you run across any problems that you feel may be common when measuring things like circles or spheres? What do you think could be done to make measuring these items more accurate?

Pick the set of measurements you made for one of the objects. Quickly (with a calculator or computer) calculate the ratio of the circumference to the diameter. What is the value? Is this value exactly ? Is the value at least close to about 3.14? Should the value be close to 3.14? Why, and if it is not, why do you think it isn’t?

We measured many different objects in the hopes of getting the same results, but undoubtedly there was variation in the values we will get for. This is called experimental error, and is something that cannot be avoided. The best we can hope to do is be aware of where it comes from and keep it to a minimum. What do you think were some sources of error in your measurements? How would you keep theses sources to a minimum, if you could?

In this experiment you had to measure different objects many times in the hopes of attaining the same result for each. Why was this done? Would your overall (average) measurements be more, or less accurate if you performed this experiment with twice as many objects? Explain.

Looking ahead: Next we will take our data collected from this experiment and analyze it using computers to get a result for the value of .