Proof that A=πr2
Start by dividing a circle into sixteen sections like an orange. We know that together they add up to the area of the whole circle.
Now take all the green sections and line them up next to each other like the bottom teeth of some wild animal. On top of them, line up all the orange pieces like the top teeth of the animal, and then fit them together as if the animal had closed his mouth.
All the teeth together look almost like a rectangle. The short side of this rectangle is the radius of the circle. The long side of the rectangle is half of the circumference of the circle, or 2πr. If we multiply them together to get the area of the rectangle, we get r (πr) or πr-squared - the area of a circle.
If we cut our circle into smaller sections, our rectangle will be straighter, but this is enough to see what it would be like.