How to convince a layperson that the $\pi = 4$ proof is wrong? The infamous "$\pi = 4$" proof was already discussed here:
Is value of $\pi = 4$?
And I have read all the answers, yet I think that they will not be of much help to me if I try to explain this thing to a non mathematician. The main missing point, in my opinion, is the fact that length of curves is defined using polygonal approximations (discrete approximation of the curve obtained by taking the straight-lines connecting a finite sequence of points on the curve). 
However, a layman would ask "why is your strange 'polygonal approximation' method correct, but the $\pi = 4$ proof's method incorrect?" and I have to admit I fail to see strong arguments to convince him here.
So my question might be better stated as "convince a layman the correct way to measure lengths of curves is our (the mathematician's) way"; however, I'm interested specifically in the $\pi = 4$ proof and will be glad to hear totally different approaches to it.
 A: The way I like to think of it is that while the length contained in each peak DOES decrease, there are more peaks each time, so the total length stays the same. Or more colloquially, it gets (infinitely) fuzzy but it never gets smooth.
A: Use a similar "zig-zag" approach to "show" that the diagonal of a $100$ meter by $100$ meter field is $200$.  Everybody who has ever crossed a field will know that walking $1$ meter north, then $1$ meter east, then $1$ north, then $1$ east, and so on is a lousy way to do it.
A: I think the problem is that that there is a mix-up between perimeter and surface. 
By caving in the corners, the diagram gets people's attention on the apparent similarity in surface of the erstwhile square and forget that what matters is the perimeter. 

Show him this picture and ask him which line is longer. Then ask him if breaking the red line with even more corners would make it shorter.
Once he realize that it is the length of the line that matters and not the area, the argument should be understood.
A: Recently, I had the occasion to explain this to someone. Here's how I went about it. Lets consider where the case of the diagonal of a unit square seems to be well approximated by the step-like curve. At first sight, it seems convincing simply because as the steps increase the jagged line seems to become like a straight line. But this is simply because we can only see up to a certain resolution. Whatever be the number of steps, if one were to zoom into a part of the figure, we'd see exactly a similar sight as before we zoomed in. This is reflected in the fact that in the limit the length of the step-like curve is still $2$. At, this point the person will say but the enclosed figure seems to look the same. And then you say, Exactly! And the measure of the enclosed planar figure is called Area, which indeed tends to the area of the rectangle (this is the basic intuition behind basic integration).
Added
On hindsight, its better to say that the enclosed figure is the one formed by the steps and the diagonal. And the fact that they look the same means that the measure of the enclosed figure (area) tends to zero.    
A: If you zig-zag instead of following a straight line, your mileage will be higher than if you'd gone straight.  A similar thing happens here.
A: Show them that if you did a $\pi$ approximation by using the area of the figures that the approximation does approach the actual value of $\pi$. You can then tell them that while the area of the figure is approaching the area of the circle the perimeter doesn't approach the circumference, because of the way that it was cut.
You could then talk about how if you circumscribed regular polygons with an increasing number of sides, that the perimeter WOULD approach the circumference.
I think that would do a pretty decent job.
A: As I understand it, one rigorous resolution to this "paradox" comes about by noting that two limiting processes are involved: measuring length requires a limit process (i.e. finer and finer approximations to the curve by discrete polygons), and the square $\rightarrow$ circle construction also requires taking a limit.  Depending on the order in which these limits are taken, you can get either $4$ or $\pi$.
In answer to the following question:

However, a layman would ask "why is your strange 'polygonal approximation' method correct, but the π=4 proof's method incorrect?"

Perhaps one approach might be to show that other intuitive methods of measuring length also give the same conclusions: for example, one method for measuring the perimeter of a shape is to trace its outline in string, and then measure the length of the string.  In the case of a real, physical "string", this method is limited in accuracy by maximum curvature that can be attained with the string.  If the shape involves too many tight zig-zags then those will be smoothed out leading to an underestimate of the total perimeter.  Finding the actual perimeter of the shape involves moving to finer and finer types of string (from thick rope, to cotton twine, to thread, to...), which can be forced into tighter and tighter curves.
In other words, we again have a limit process, but here it involves the maximum curvature of the approximation, rather than the number of segments in a polygon.  This leads to exactly the same results: depending on the order in which the limits are taken (i.e. in the curvature of the string, or in the construction of the circle), we get either $4$ or $\pi$.
A: The simplest way to convey what is wrong with this "proof" that I can think of is:


*

*As you subdivide the square to closer approximate the circle, repeatedly "to infinity", the zig-zags in the line not only become infinitely small, but at the same time you end up with an infinitely large number of them.  You cannot have one effect without the other.

*Ask the person "How big is an infinitely large number of infinitely small things?"
Hopefully this will manage to convey the notion that just because something is "infinitely small", doesn't mean it is literally zero, and that an infinitely large number of them doesn't actually tell us anything more about a problem than "the result is multiplied by an unknown number between zero and infinity"
A: Non-rigorous proof
Give this one a try - I think the part "the author has convinced you that the shortest distance between two points is not a straight line" bit will be easily understood / accessible, even if the mathematics is not.
Also the "staircase" analogy might help, most people will understand that.
A: Tell your layman about the difference between the euclidean metric and the "taxicab metric". The euclidean length of a segment $\Delta {\bf z}=(\Delta x, \Delta y)$ is $\sqrt{\Delta x^2 + \Delta y^2}$ whereas the "taxicab" length of this segment is $|\Delta x|+|\Delta y|$. "In the limit" this implies that the euclidean circumference of the unit circle is $2\pi$, whereas the "taxicab circumference" is $4$.
A: Disclaimer This is not a rigorous approach, at all, since the question is not framed as it was in the similar, earlier, linked question.  I'm trying to provide an appeal to intuition that helps to "get one's foot in the door" when trying to convince a laywoman that it is faulty. Once such an individual comes to doubt the faulty proof, he/she will be better prepared, and open to, a more rigorous explanation/proof that the circumference of said circle is, indeed, $\pi$ 

In trying to keep to the "perimeter of square = circumference of circle inscribed in the square" conjecture, but in keeping with user6312's suggestion: for those who believe the "proof" you linked to, and even those who aren't sure what to believe... 
Ask them if they were in a race, and had the choice of whether to run on one of two tracks: 
1. a square track of perimeter 400 meters, or 
2. a circular track that is "inside" the square, touching the square only at the midpoints of the square's edges, which track would they choose?
Most, I suspect, would opt, rather immediately, to choose running on the circular track . Upon their response, ask them "Why? - then show the connection to the "proof that $\pi = 4$", and "run" through the mathematical approach to arriving at the circumference of a circle (no pun intended!).
Alternatively, you could ask them to bet on a race where two world-class top-seeded sprinters were racing against one another, with Runner1 randomly selected to run the race on the "square track", and Runner2 to run on the "circular track." On whom would they bet, and why? Again, you'll have your "foot in the door" to challenging the "$\pi = 4$" claim.
Just a thought...
A: Notice that in this argument $\pi$ ends up equalling 4 since that is the perimeter of the circumscribing square. Taking a square is completely arbitrary though, just show your layman the same argument starting with a triangle or pentagon or even something like this:

Then after showing about 3 separate cases all ending up with the circumference of the starting figure instead of $\pi$, I'd say even the least mathematically inclined should get the invalidity of the argument.
A: I would explain as follows:
Say to the layman that if you added up the length of all the horizontal lines without the vertical lines then the result would equal the circumference of the circle. I am not saying that this is correct but it would sound  correct and logical. It would then be clear (even if still confusing) that adding up all the horizontal and vertical lines will give an incorrect answer. He would intuitively understand that the answer should be less than 4.  
A: Think of it this way: when you use the method given above, the perimeter never approaches that of a circle, for the perimeter is always the same (meaning that you cannot apply the concept of 'approaching' something...all you approach is an infinitesimally small set of zig-zagged lines, and even more mess than you started with :() Now, in contrary, when you apply the same method to calculate the area of the circle and not the perimeter, you definitely do approach something, as you continue the process for infinitely many times -- for the area gets increasingly shorter and shorter -- and the limit is the area of the circle. I understand that it can be argued that there still are many zig-zagged lines when you approach the area, but the point is that it can always be made shorter by continuing the process again by many times, which will give a shorter area each time but never an area smaller than that of the circle, which is the definition of 'approaching' a limit. However, in the case of perimeter, all the processes you do are in vain as you never really approach any value, which is to say that its limit does not exist.
A: pi can be experimentaly proven to differ significantly from 4,  eg: use a piece of string and a cylindrical can.
considering this demonstration in the context of the faulty proof should quickly lead to an understanding of flaws of the rectilinear approximation of the circumference.
A: The most direct explanation is that the method given is one way to put an upper bound on the circumference, but to get the smallest upper bound for the circumference you need to consider every possible way and pick the smallest.
If you do that, you quickly see there are shorter polygons that also bound the circumference.
And if you look at the shortest possible polygons you can construct, you in fact converge to .
