Why does squaring the radius of a circle times pi equal the area? (I.e., why does the area of circle representable as $\pi r^2$ ?)What is the relationship between the radius of a circle and the area?
4 Answers
I prefer the following illustration.
"CircleArea" by Jim.belk - Own work. Licensed under Public Domain via Wikimedia Commons.
Essentially, you need to square some linear measure of the circle to get to an area. The radius and the circumference are in any case related by a constant - $2\pi$ - and it happens that squaring the radius gives the simplest constant to obtain the area.
There is a pretty visual of how the area and radius are related here.
if you assume that $\pi$ is the ratio of the diameter of the circle to the circumference, then you can divide the circumference, say, into $1000$ points at equal distance. now join each of these segments to the center to make a small pie. arrange the pieces so that they make an almost rectangle of length $\pi \times \text {radius}$ and width the radius of the circle. the area of this rectangle is $\pi$ times the radius squared.
