Why Is Pi the Same in C = 2 Pi r 
and A = Pi r^2?

We slice up a circle like a pie and rearrange the sectors to form a figure that is roughly a rectangle. As the sectors become thinner, the rectangle's left and right sides become more vertical and the top and bottom become flatter and more horizontal. We can imagine that in the limit we get an actual rectangle.

The height of the rectangle equals the radius of the circle, r, and its length C/2. This is because the top and bottom of the rectangle are made up of all the small arcs that add up to the circumference of the circle. Since the rectangle has the same area as the circle, the length times the height is C/2 r = Pi r^2. Canceling r on both sides and multiplying by 2 gives the formula C = 2 Pi r.

PiRectangle[4];

PiRectangle[6];

PiRectangle[8];

PiRectangle[10];

PiRectangle[12];

PiRectangle[20];

PiRectangle[30];

PiRectangle[50];

Implementation

SetOptions[Graphics, AspectRatio -> Automatic];

Needs["Graphics`Colors`"]

Usage messages

PiRectangle::usage = "PiRectangle[n] draws a circle cut 
into n equal sectors (n needs to be even) and rearranges 
them to form an approximate rectangle with dimensions 
1 X Pi.";

Function definitions

PiRectangle[n_?OddQ] := 
	Print["PiRectangle needs an even argument."]

PiRectangle[n_?EvenQ] := Module[
	{a, x, y},
	a = 2 N[Pi] / n;
	x = 2 Sin[a / 2];
	y = - Cos[a / 2];
	Show @ Graphics @ { 
		Red, Table[
			Disk[{-2, 0}, 1, {2 i, 2 i + 1} a], 
			{i, 0, n - 1}
		],
		Green, Table[
			Disk[{-2, 0}, 1, {2 i + 1, 2 i + 2} a], 
			{i, 0, n - 1}
		], 
		Red, Table[
			Disk[{i x, 0}, 1, {-a/2, a/2}- N[Pi/2]], 
			{i, 0, (n - 1) / 2}
		],
		Green, Table[
			Disk[{x/2 + i x, y}, 1, {-a/2, a/2} + N[Pi/2]], 
			{i, 0, (n - 1) / 2}
		]
	}
]