I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below:
The area of a square is like a line, the height (one dimension, length) placed several times next to each other up to the square all the way until the square length thus we have height x length for the area.
The area of a circle could be thought of a line (The radius) placed next to each other several times enough to make up a circle. Given that circumference of a circle is $2 \pi r$ we would, by the same reasoning as above, have $2 \pi r^2$. Where is the problem with this reasoning?
Lines placed next to each other would only go straight like a rectangle so you'd have to spread them apart in one of the ends to be able to make up a circle so I believe the problem is there somewhere. Could anybody explain the issue in the reasoning above?
Best Answer
The main issue is that you don't form an area by placing lines next to each other--you need to place strips next to each other. As you say, to form an $a\times a$ square, you can place $n$ strips of dimension $a\times w$ next to each other, where $w=a/n$, giving total area $naw=a^2$.
Your suggestion amounts to forming the area of a circle of radius $r$ by placing $n$ strips of dimension $w\times r$ next to each other radially, where $w=2\pi r/n$, giving area $nrw=2\pi r^2$. But look what happens if you do this:
The problem is that the strips overlap, so the total area of the $n$ strips is greater than the area of the circle. If you run the animation (by reloading the page, if necessary) you can convince yourself that in the $n\rightarrow\infty$ limit, half of each strip contributes to the final area. (Observe that as strips are added in the counterclockwise direction, roughly half of each strip gets covered by subsequent strips.)
We can fix the problem of overlapping strips more easily by using triangles of base $w$ and height $r$ instead of rectangles:
This gives area $\frac{1}{2}nrw=\pi r^2$.