Here's a quick and dirty proof with some black magic...
Put the unit circle with radius 1 inside the square with sides measures 2.
Now start throwing darts at it. If you take the proportion of the darts that land
inside the circle, you will get a ratio that is approximately equal to pi.
To see this,draw 4 radius so that they are perpendicular to each side of the square.
There are four sections created. In each case, the area of the 1/4 circle is pi*1^2/ 4 = pi/4 while
the are of each unit square is 1. So the ratio is pi*4/4
The total ratio is then pi...
Now the right triangle created by these radii form 4 congruent right triangles with legs 1 and 1.
The hypotenuse is then sqrt(2). Since there are four of them, the perimeter of these hypotni are 4 * sqrt(2)
Meanwhile the circumference is pi * 2= 2 * pi.
Now here is where the fun begins.. the square's is used to approximate the circle's circumference.
So we are approximating 4 * sqrt(2) vs. 2 * pi
We can go so far as to use the APPROXIMATAELY EQUAL symbol here, to denote this approximation.
4 * sqrt(2) vs. 2 * pi
2 * sqrt(2) vs pi <--- divides both sides by 2
sqrt(2) + sqrt(2) vs pi <---- rewrites left side as 2x = x +x where x = sqrt(2)
Now note that the right triangles UNDERESTIMATE the actual area and perimeter of the circle.
So we can improve the approximation by "tweaking" one of the square roots. Here is where the
"black magic" come into play
sqrt(2) < sqrt(3) < pi/2
sqrt(2) + sqrt(2) < sqrt(2) + sqrt(3) < pi/2 + pi2 = pi
which SUGGESTS rather than proves the approximation.
I have uploaded a rather simple diagram that illustrates the construction.
The filename is pi approx sqrt2 + sqrt3.jpg