Mathemagic Task: discovery of "pi"

TASK

To find a relation between circumference of a circle to its radius

I am using a bangle, a  standard compact disc and a mini compact disc  as my three circular objects, rulers with inches and centimeters, and some string, to derive a relationship between the diameter of a circle and its circumference.

1.      I have bangle. Measuring the diameter and circumference of bangle, I found its

Diameter = 2.3 inches or 5.8cm

And

Circumference = 7.3 inches or 18.3cm

          The ratio of circumference to its diameter = 3.17391… or 3.15517…

         

2.      I have standard compact disc. Measuring the diameter and circumference of this, I

Found its

Diameter = 4.7 inches  or  12.1cm

And

Circumference =  15 inches  or 38.1 cm

                The ratio of circumference to its diameter = 3.19148… or 3.14876…

3.      I have mini compact disc. Measuring the diameter and circumference of this, I

Found its

Diameter = 3.1 inches  or  7.9 cm

And

Circumference = 9.7 inches  or  25 cm

                The ratio of circumference to its diameter = 3.12903… or 3.16455…

summary

This is a great task to find a relationship between diameter of a circle and its circumference. And also this task is based on the “discovery of pi(π)”. Actually, we also discovered pi by the help of this task.

The ratio of circumference to its diameter gives us a interesting value, as we can see above in our experiment.

We can observe that this ratio is non-terminating and non-repeating.

Hence we can say that this is an irrational quantity.

A standard value of pi = 3.14159…

Here is step by step processor that I made through my experiment.

A.    Problem solving process:

1.      Measurement tools used:

I used a bangle, a  standard compact disc and a mini compact disc  as my three circular objects, rulers with inches and centimeters to measure diameter, and some string to measure circumference.

Main point:

while measuring the diameter for any object, this is necessary the line of measurement must passes through its centre. Ex: when I need to find out the diameter for my bangle. First I need to get its centre. Because diameter always passes through centre. To get this, put one  ruler across the bangle anywhere, and then by the reading, find its mid point. Then put one string across the bangle, passing through and perpendicular to this mid point. This string will be the line of measurement. And by now  we can measure diameter.

2.      Data collection process:

I collected my data in both metric(centimeters) and    traditional(inches) units.

                     A standard relation between centimeters and inches is given by,

1 inch = 2.54 centimeters(cm)

3.      Measurments are approximation:

There are two main reasons that’s why we can say that our measurments are approximations.       

(a)   While measuring diameter, sometimes not necessary the line of measurement passes through its centre exactly. That thing may create an error. And that’s why we consider our measurments as approximations.

(b)   While measuring circumference, bending of string at its parts may not give an accuracy. That cause an error too. And that’s why we consider our measurments as approximations.

4.      A table of the data collected for all three items:

objects	Measurement units	Diameter	Circumference	Ratio of circumference to its diameter
1. a bangle	Inches	2.3	7.2	3.17391…
	Cm	5.8	18.2	3.15517…
2. a standard compact disc	Inches	4.7	15	3.19148…
	Cm	12.1	38.1	3.14876…
3. a mini compact disc	Inches	3.1	9.7	3.12903…
	Cm	7.9	25	3.16455…

For example,

when we measure the diameter and circumference of bangle in inches then get the ratio 3.17391…

And when measure in cm then get the ratio 3.15517…

Hence we can observe that different units affect the precision.

B.   The use of collected data to derive an experimental value of pi:

The ratio of the circumference to its diameter gives us pi.

Circumference/Diameter = pi(π)

For every collected data ,the ratio is also shown in the table above.

C.     Ananlysis of the degree of error:

1.      Degree of error in measurments and experimental value of pi, using the known value of pi(3.14159…)

The percentage of error is used to evaluate the degree of error.

Percentage of error = ({experimental value - actual value)/actual value} *100

From the first observation, when (units in inches)

            Percentage of error ={ (3.17391… -  3.14159…)/3.14159} /*100

            Hence

            Percentage of error = 1.03%

From the first observation, when (units in centimeters)

            Percentage of error={(3.15517… - 3.14159…)/3.14159…}*100

            Hence

            Percentage of error = 0.4%

Here we can observe that the percentage of error is less , while measuring in centimeters.