We start by dividing a circle with radius $r$ into a number of equally sized circle sectors. We can then distinguish the top and bottom half of the circle by filling them with different colors. Because the circumference of a circle is $2\pi r$ the arc length of each semicircle is half of this, which is $\pi r$.

Now, imagine that we "unfold" all of the circle sectors. We place the blue part as teeth pointing downward and the green part as teeth pointing upwards. We put these two together which gives us a figure that looks like a parallelogram. As we haven't removed or added anything, the area of the figure below is the same as the circle's area.

We can't easily calculate the area of this figure, but note that if we divide the circle into more, even smaller circle sectors, then the figure will begin to look like a rectangle more and more. Here, for example, we divide the circle into twice as many circle sectors as above.

The vertical sides becomes more vertical, and the horizontal sides becomes more horizontal. If we divide the circle into infinitely small circle sectors, the figure will become a perfect rectangle, with base $\pi r$ and height $r$. The area can then be calculated as which is the circle's area.