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?
$\endgroup$24 Answers
$\begingroup$I prefer the following illustration.
"CircleArea" by Jim.belk - Own work. Licensed under Public Domain via Wikimedia Commons.
Check this: Calculus proof for the area of a circle
$\endgroup$$\begingroup$Essentially, you need to square some linear measure of the circle to get to an area. The radius and the circumference are in any case related by a constant - $2\pi$ - and it happens that squaring the radius gives the simplest constant to obtain the area.
There is a pretty visual of how the area and radius are related here.
$\endgroup$$\begingroup$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.
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