How to derive the number pi = π = 3.1415927...


The number pi (π) is equal to 3.1415927 seven places after the decimal point.

It is commonly used when calculating the circumference and the area of a circle, which formulas are equal to

2 × π × r and π × r2    respectively

Ever wondered where pi came from or why pi is equal to 3.1415927?

You can derive the number pi by doing the following experiment:

Take any piece of string. Use a ruler to measure the length of the string in inches or centimeters or any units you want


String and ruler
Write down the length of the string on a piece of paper.

This length is also the perimeter or circumference of the circle after you make a circle with the string

Now play with the string until you make a circle with it. Then, use a ruler to measure the diameter of the circle.


Circle made with string
Write down the length of the diameter on a piece of paper.

Do a ratio of the circumference to the diameter

In other words, divide the length of the string by the diameter.

You should find something very close to 3.1415927.

The piece of string you see on the picture above has a length of about 22 inches and I found a diameter of 7 inches. 22 divided by 7 is equal to 3.142857143.

It is not equal exactly to 3.1415927, but it is ok because in practice we use 3.14 to do calculation.

All we needed here was a good approximation.

Repeat the experiment above with a longer or a shorter piece of string.

You should find a similar answer for pi because the size for the circle does not matter.

In general, regardless of the size of the circle, the following is always true.


Ratio of circumference to diameter
C1 / D1
   =  
C2 / D2
   =  
C3 / D3
   =  
C / D
  =   3.1415  =   π
C1 / D1
   =  
C2 / D2
   =  
C3 / D3
   =  
C / D
  =   3.1415  =   π



If
12 / 3
= 4 , then 12 = 4 × 3


If
C / D
= π, then C = π × D


Since D = 2 × r, the formula above can also be written as C = 2 × π × r





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