Area of a Circle
First Method: Rearranging Sectors
If we divide a circle with radius into sectors of equal size, we can rearrange them in the pattern shown at the right. At first, this does not seem to be very helpful, but notice what happens as we increase the number of sectors.
What shape does this figure resemble as ?
What does the height, , approach as ??
As , what does the base, , approach?
As , what does the area of the figure approach?
Second Method: Inscribed Regular Polygon
In the diagram notice a few relationships:
Looking at the right triangle we have , which means the apothem has length:
If is the side length of the regular polygon, then we also have , so we can express the side length as:
Therefore, the perimeter is:
Regarding the angle , we have two copies of the central angle corresponding to each of the sides of the polygon, and all of these add up to , so .
As , what does the length of the apothem approach?
As , what does the perimeter approach?
Therefore, as , what does the area approach?