Pi: Facts and Figures
---------   =   3 times [(r2)/(r3)] = 3 / r.
(4/3)πr3
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
  +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 36283185840707964601769911504424778761061946902654 867256637168)  4703/33215 = 0.1415926539214210447087159 37741629/266550757 =               0.14159265358980015952458915732886100938741659623198                89416033434862839275297950101113387571433533764077                81126654237939380528564771624340200241862378203637                96603286330190388466989047042923986143472104264179                59863456699918507453347806474246854230468383175516                54899258080141186768417243699668052340215263391654                89595664513532032475150689592676714851723343633197                78529085175323662652363054440697011413852409355566                00201289242633814767218987864288826611736090473755                43562984516303587162575569050062761592532262063693                9327,                    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.