Why is the Area of a Circle r2?

 

Here is a circle:

Circumference = * diameter

Below, the "slices" of the circle are reorganized to make a nearly four-sided figure. To do this, the 8th slice is divided in half and one half is placed at each end of the figure:

If the slices of the circle kept getting smaller and smaller, the number of slices would get bigger and bigger. Also, the length of the rectangle made out of the slices would get closer and closer to being a straight line equal to half the circumference of the circle.

As the number of slices approaches infinity, the following equivalencies get closer and closer to true, and the area of the rectangle becomes a better and better approximation of the area of the circle. Thus, we could use the formula for the area of a rectangle to calculate the area of the circle:

Length of rectangle = 1/2 circumference of circle

Length = 1/2 ( * diameter)

Length = 1/2 ( * 2 * radius)

Length = * radius

Width= radius

Area = length * width

Area = * radius * radius

Area = r2