Area of a circle

Question. Why is the area of a circle equal to $\pi r^2$ ?

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 $2\pi r$ , so the length of the top edge of the approximate rectangle is $\pi r$ .

$A \approx \pi r^2$

The height of the rectangle is roughly $r$ , and hence the area of this region is roughly $\pi r^2$ .

The larger the number of pieces, the better the approximation to a rectangle. The limiting rectangle has base $\pi r$ and height $r$ , so the area of this rectangle is exactly $\pi r^2$ . The area of this rectangle is equal to the area of the circle, so the circle also has area $\pi r^2$ .

In the following figure, the yellow region changes from a circular region to a rectangular region of the same area.

$A = \pi r^2$

M	T	W	T	F	S	S
1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30

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