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(Pi): Facts and Figures
 Edited by The Starman

Pi () to 100 decimal places is:

      3. 14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679

A  Circle  of   (Pi)

  is the ratio of a circle's circumference to its diameter (π = c / d) which also means that the circumference of a circle is Pi times its diameter (c = πd) or twice Pi times its radius (c = 2πr). If we make the diameter 1 unit, then its circumference will equal π units. The area (A) inside a circle is Pi times the radius squared or A = π r2 (see links below for proof *). If a circle has a diameter of 2 units, then its area will equal π units squared; for example, a circle of 2 cm in diameter has an area of π cm2.  

For various proofs that the area A = π r2, see: Area of a Disk, Archimedes on C & A and this Video.

is the 16th letter of the Greek alphabet (it also denoted the number 80 in ancient Greece). Note that the pronunciation of this letter in Greek is like the English word 'Pea' (the same way they say the name of the letter 'P') or perhaps like the p and i in the word 'Pit.' But it's NEVER pronounced like the English word 'Pie' in Greece! To create that type of sound, Greek might use the diphthong 'ai' (diphtong means two vowels together; like the 'oi' in the English word 'oil'); so, 'pai' in Greek might sound like 'pie' in English.

So, how did this Greek letter become the Mathematical symbol it's known for today?
In the mid to late 1600's, some mathematicians were using (a lowercase Pi divided by a lowercase Greek Delta) to note the perimeter divided by the diameter of a circle; 3.14159... . The symbol, π, was used in this manner because it's the first letter of the Greek words, perifereia (periphery) and perimetroV (or perimeter; which could also mean the circumference of a circle). Many geometrical terms in English come from Greek, including of course, geometry itself, gewmetria (gew = earth + metria = measure).

But William Jones is often cited as being the first author to use the Greek letter π for this constant in his 1706 work, A New Introduction to Mathematics.

Spin that Circle into a Sphere

    A circle has the smallest perimeter length (its circumference) to area ratio (for a given shape), which is: (2πr)/(πr2) = 2 / r.
    Centuries ago when mathematicians were also philosophers, because of this smallest ratio property, they considered the circle to be the 'perfect' two-dimensional shape.
    Now, if we spin a circle around any one of its possible diameter lines, we bring everything up one dimension, and turn that circle into a 3D sphere with the property that this 3D shape also has the smallest surface area to volume ratio (for a given volume or radius); that ratio being 3 / r. Further below, we'll explain why the Surface Area of a sphere is 4πr2 and its volume is (4/3)πr3, but for now, take a look at how that ratio results in 3/r:

That's pretty cool, how the 2D ratio for the circle turned out to be 2/r, and the 3D ratio for the sphere was 3/r.

Another curious ratio is that of a sphere's surface area to the area of the circle we created it from, which is exactly 4: The surface area of a sphere being 4 times πr2 (the area of its circle).

There are some really nice and somewhat simple ways (compared to using Calculus) to show how the formulas for the Surface Area and Volume of a Sphere can be derived!

The Volume is easier to show, if we simply assume for now that we have already proved the Surface Area, so let's start with that. Next, think of the whole surface of a sphere as being made up of a huge number of little squares and that each of those squares is the bottom of a pyramid that has of course as its height, the radius of the sphere. We will also assume here that the Volume of a Square Pyramid is one third (1/3) times its perpendicular height from the middle of the Base times the Surface Area of that Square Base, or: (1/3) x BaseArea x height, or in the case of our Sphere: (1/3) x BA x r. But, since the sum of all those tiny pyramids bases is equal to the Surface Area of the Sphere, then its Volume would be equal to: (r/3) x Surface Area = (r/3) x 4πr2 = (4/3)πr3.

We have, of course, ignored the fact that each of those little "square pryamids" would have a slightly upward curved base that is not exactly a completely flat square! But the smaller we make those bases, the flatter that tiny square becomes and we arrive at values where it makes no practical difference. So that's why we have simply showed you a way to arrive at the formula, rather than having actually proved what it is; which can only be done using the principles of Calculus.

The formula for the Surface Area of a Sphere can be shown as follows:

Rather than take the time (I'm getting way too little sleep already!), There's a nice presentation of this formula here, or watch This VIDEO.

Fractions approaching Pi

Fraction Decimal Approximation Percent Deviation from Pi
25/8 = 3 + 1/8   3.125 (exactly)   -0.52816 %
22/7 = 3 + 1/7   3.142857142857   +0.04025 %
333 / 106   3.14150943396   -0.00264896 %
355 / 113   3.14159292035   +0.00000849 %
104348 / 33215   3.14159265392   +0.00000001 %
837393900 / 266550757   3.14159265358980   +2.2 x 10^(-13) %

The following lists the repeating decimal portion (underlined) of the fractions above:

   333/106 = 3 + (15/106 = 0.141509433962264)

   355/113 = 3 +
   (16/113 = 0.14159292035398230088495575221238938053097345132743

4703/33215 = 0.1415926539214210447087159

37741629/266550757 =

                   and even more digits before finally repeating!

The Babylonians were the first to use (25 / 8)

The fraction (22 / 7) is probably the most famous approximation for Pi (about 3.142857). It's actually composed of 3 + (1 / 7) and the fractional part (1/7) is famous in its own right as being the first fraction to produce a 'repeating decimal' number with more than a single digit! (1/7 = 0.142857 142857 142857 ... etc.)

(355 / 113) is about 3.141592920354 and only 0.00 00 0849% larger than Pi.

If one were to find the circumference of a circle the size of the known universe, requiring that the circumference be accurate to within the radius of one proton only 39 decimal places of Pi would be necessary!

Some Historical Facts about Pi
(in Chronological Order)

The Babylonians are credited as having recorded the first value for Pi (around 2000 BCE) and they used (25/8).

The earliest known reference to Pi is on a Middle Kingdom papyrus scroll, written around 1650 BC by Ahmes the scribe.

In around 200 BC Archimedes found that Pi was between (223/71) and (22/7). His error was no more than 0.008227 %. He did this by approximating a circle as a 96 sided polygon.

Ludolph Van Ceulen (1540 - 1610), Germany, spent most of his life working out Pi to 35 decimal places, thus the reason, Pi is sometimes known there as Ludolph's Constant (in German: die Ludolphsche Zahl).

In 1706 (a fruitful year for Pi), Machin found this rapidly converging formula for its calculation:
    π/4 = [4 x arctan (1/5)] - arctan (1/239).

Also in 1706, the Welshman, William Jones, became the first person to use the Greek letter π as an abbreviation for the periphery of a circle with unit diameter. Euler adopted the symbol, and it quickly became the standard notation for Pi in all mathematical contexts.

Some Basic facts about π
(including its Irrationality and Transcendence)

In 1768 (or 1761), Johann H. Lambert proved Pi is irrational. An irrational number is a number that cannot be expressed in the form (a / b) where a and b are integers. Some other famous irrational numbers include the square root of 2 (the square root of any prime number is irrational) and e.

In 1882 Ferdinand Lindemann, proved the transcendence of Pi.

Transcendental in mathematics can mean: Not capable of being determined by any combination of a finite number of equations with rational integral coefficients. Another way of stating a number is 'transcendental' is to say it can never be the solution to any finite polynomial (that is, lots of numbers added in a series) with whole number coefficients.  This is why it is impossible to "square the circle".

It is easy to prove that if you have a circle that fits exactly inside a square, then π = 4 x [(Area of circle) / (Area of square)] because each side of the square is equal to the diameter of the circle, or 2 times the radius:

π = 4 x [ (πr2) / (side2) ] = 4 x [(πr2)/(4r2)] = (πr2)/(r2) = π.