Interesting Facts about Pi

You should have noticed that the ratios for all the circles are very close.  The values that you should have gotten should be very close to 3.1. The measurements that you did were not that precise.  If they had been, then all of the ratios would be extremely close to each other.

What that tells us is that there is a fundamental constant¹ that works with every single circle.  The name of this constant is pi and its value is close to 3.1415926535897932... The Greek letter is used  to represent this important constant. Click here to see with five hundred digits of precision.

One more thing that you may want to do with your data is get the mean or average² of all the ratios (or approximations of ) that you measured to see how close you are to the value of in the previous paragraph.

In the rest of this page we have a few interesting facts about .

 

	, a fundamental constant of nature, is one of the most famous and most remarkable numbers you have ever met.
	The Egyptians and the Babylonians are the first cultures that discovered about 4,000 years ago. Here is a small table that shows some of the very old discoveries of : Culture/Person Approximate Time Value Used Babylonians 2000 BC 3 + 1/8 = 3.125 Egyptians 2000 BC 3.16045 China 1200 BC 3 Bible mentions it3 550 BC 3 Archimedes 250 BC 3.1418 Hon Han Shu 130 sqrt (10) = 3.1622 Ptolemy 150 3.14166 Etc...
	William Jones, a self-taught English mathematician born in Wales, is the one who selected the Greek letter for the ratio of a circle's circumference to its diameter in 1706.
	is an irrational number. That means that it can not written as the ratio of two integer numbers. For example, the ratio 22/7 is a popular one used for but it is only an approximation which equals about 3.142857143... Another more precise ratio is 355/113 which results in 3.14159292...  This was given to me by a student. Another characteristic of as an irrational number is the fact that it takes an infinite number of digits to give its exact value, i.e. you can never get to the end of it.
	Since 4,000 years ago and up until this very day, people have been trying to get more and more accurate values for pi. Presently supercomputers are used to find the value of with as many digits as possible. Pi has been calculated with a precision containing more than one billion digits, i.e., more that 1,000,000,000 digits!
	Here are three different ways to approximate the value of : / 2 ~= (2*2*4*4*6*6*8*8*...) / (1*3*3*5*5*7*7*9*...) / 4 ~= 1 -1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... ~= 3 + 1/10 + 4/102 + 1/103 + 5/104 + 9/105 + ... The symbol "~=" means approximate. They are not equalities but can be very close. Try them out in your calculator! It is fun!
	If we use the variables C for the circumference and d for the diameter, then can be expressed as the following ratio: = C / d This formula can be solved to find the value of d: d = C / and the value of C: If you were given the diameter of a circle, could you find out its circumference? And what if you were given the circumference? Could you find out what the diameter is?
	appears in many different formulas that have nothing to do circles.
	Euclid of Alexandria (325 - 265 BC) is the one who proved that the ratio of C over d is always the same, regardless of the size of the circle. He did it by inscribing regular polygons (i.e., an octagon or 8-sided figure) inside circles of different sizes. He was able to show that the perimeter of the polygon was proportional to the radius (which is half of the diameter), regardless of its size. He then increased the number of sides of the polygon, realizing that as he increased them, the perimeter of the polygon got closer and closer to that of the circle. Therefore, he was able to prove that the perimeter of the circle, or circumference, is proportional to the radius and also to the diameter.
	Archimedes of Syracuse, Sicily (287 - 212 BC) did the first theoretical calculation of . He used methods similar to the ones used by Euclid by inscribing a regular polygon inside a circle. He started with a hexagon and then polygons of 12, 24, 48, and finally 96 sides. He also used one of Euclid's theorems to develop a numerical method for calculating the perimeter of the polygons. Archimedes obtained the approximation 223/71 < < 22/7.

[Note 1] -- A constant is a number that is fixed and doesn't change. In this case the constant is exactly the same number for all circles.

[Note 2] -- To get the average of all your ratios what you need to do is add them all together and divide the result by the number of ratios that you added together.

[Note 3] -- 1 Kings 7:23 reads:

He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a circumference of thirty cubits to measure around it.

Thirty divided by ten gives a value of 3. However, it is interesting to note that the word circumference happens to be spelled with an extra letter. Since in Hebrew all letters are also numbers, if we take the ratio of the value for the word as it is written (111) to the normal spelled word (106) we get the number 1.047169811... If you multiply this number by 3 you get 3.141509434... An amazingly close approximation to !

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Last updated: Tuesday, 25-Nov-2003 18:57:44 GMT

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