A simple way to visualise the πr? formula is this:
(refer to this diagram for the following explanation: http://en.wikipedia.org/wiki/Area_of...angement_proof
First we draw a circle with a radius of r. Now, divide the circle into say, 20 equal sectors like a pie chart. If you look closely, each of the 20 sectors would look similar to a triangle, except that their base is an arc, not a straight line. Now, instead of 20, cut the circle to a very, very large number of sectors ("approaching infinity" is the term we often use) - now you would have a lot of very very thin triangles. Those triangle has a very very small base, and the two remaining sides' length are r, since they are the circle's radius.
Now, if you list out the triangles in a straight line, those triangles will stretch out like a row of sharp teeth, and it will be 2πr long, since the total length of the base is the same as the circumference of the circle. Now, since you have a very very thin triangle, the side's length is essentially the same as the perpendicular height, which is the height of the "teeth".
Then, fold the "row of teeth" into half, invert one of the halves, and place it above another half. Now you would have a parallelogram, with the base of πr, and a perpendicular height of r. What is the area of that parallelogram? Base times height, which gives you πr?.
(Note that this is not mathematically stringent, but this is one of the easiest ways to explain the circle area formula, and is not too far from the strictest proof)