Is this a valid proof that the area of a circle is $\pi r^2$? We can break up the circle into an infinite amount of rings with perimeter $2\pi r$. For a given circle $r$, the outside ring has perimeter of $2\pi r$ and the smallest one has of course perimeter $0$. We can add up all the area of the infinite rings using the arithmetic series concept; we can get the average of the first term and the last term and multiply by the number of terms. Therefore, we can find the area of the circle as $\frac{2\pi r + 0}{2} \times r$ = $\pi r^2$.
I thought of this proof and was wondering if it was valid. I know the phrase "arithmetic series" to add up/integrate the rings together is not really the right wording, but is it the right idea?
 A: This would go solidly into the category of "plausibility argument" not proof. The fundamental idea could be formalized into a proof, but this goes for many(if not most) plausibility arguments. To do it properly would require lots of work to remove ambiguity from statements like "we can add up all of the infinite rings using the arithmetic series concept." You would essentially need to build up Calculus. 
A: You started off with the right idea. Area of any plane region is found by splitting into smaller regions and adding their areas. The assumption here is that the smaller regions can be chosen with familiar shapes like squares, rectangles or even triangles whose areas can be calculated using well known formulas. Naturally any shape can not be broken into an exact number of smaller regions of such shapes and hence the technique works via approximation. There are some parts of region left out when we try to divide it into a number of square shaped regions. Our hope is that the approximation has less and less error as we make finer and finer divisions.
And all this procedure can be formalized into the language of limits and techniques of integral calculus can be used to get exact values of the area of the plane region.
But your approach has a fundamental problem. You divide circle into a number of rings and you don't have the formula for area of a ring. Some people try to approximate the ring as a rectangle with circumereference as length and width of ring as width of rectangle. This approximation works well and gives area of circle correctly. But note that this is not exactly an arithmetic progression and there is no $r$ number of terms. So that kind of sum of an arithmetic progression does not make sense here.

There is another aspect here which is worth mentioning and that is definition of area of plane regions. Most ancient mathematicians never bothered to formulate an exact definition and instead regarded it as obvious that any plane region has an area which can be found by the technique of subdivision into smaller regions. However it is important to distinguish definition of area with methods of its calculation. I have provided one definition of area in this answer which you may find interesting.
A: For a slightly more detailed argument (not rigorous yet, but closer), consider a large integer radius $r$ and accumulate the contributions of $r$ rings $1$ unit large.
$$A\approx\sum_{s=0}^{r-1}2\pi\left(s+\frac12\right)=2\pi\frac{(r-1)r+r}2=\pi r^2$$
by the triangular numbers formula. Then by similarity, the formula extends to arbitrary radii.
But still remains to prove that the area of a ring is well approximated by its perimeter, i.e. that you can "straighten" it...
