Question. Why is the area of a circle equal to 
?
Answer. The first step is to look at a circle and cut it into 6 pie-shaped pieces. These pieces can be rearranged into something that looks similar to a rectangle.
Now consider what happens if you cut a circle into a larger number of small pie-shaped pieces. Here is what you get with 20 pieces of pie.
Take the pie apart and put it back together to get something that looks a bit more like a rectangle:
What happens when the pie is cut into an even larger number pieces?
Rearrange the pieces to form something that looks even more like a rectangle. The circumference of the circle is equal to 
, so the length of the top edge of the approximate rectangle is 
.
The height of the rectangle is roughly 
, and hence the area of this region is roughly .
The larger the number of pieces, the better the approximation to a rectangle. The limiting rectangle has base and height , so the area of this rectangle is exactly . The area of this rectangle is equal to the area of the circle, so the circle also has area .
In the following figure, the yellow region changes from a circular region to a rectangular region of the same area.
