Why is the area of a circle not $2\pi r^2?$ (a fake proof)

Question

So, I realise that the area of a circle is actually $\pi r^2,$ so basically half of $2\pi r^2,$ however, after trying to prove the area of a circle for myself, I came up with a really convincing and intuitive theorem for why it is 2πr^2 (even though it is not).

By the way, I've set the radius to 100/2π so that the circumference would be 100, as I wanted to mark the percentage moved across the circumference for any given angle. There is no reason for this with respects to (my attempt of) this so-called 'proof', it is just for convenience.

If you take the circumference of a circle and you flatten it out into a straight line, it would look something like this:

From here, I reason that a line segment can be drawn across any, indeed every point along the line equal to the radius of the circle, since said line is equal to the circumference of the circle to begin with, and thus this step just falls out of the definition for how a circle is even defined. Here are a few examples:

If you did this for every point across the circumfence and its corresdonding line, you would have filled in the area of the circle on the left, and created a rectangle on the right, whose length (x-axis) is the circumference of the circle and whose height (y-axis) is the radius:

It's not obvious what the circumference multiplied by the radius would be, but since we know that the diameter goes into the circumference π times, we can just rewrite the circumference as π•diameter, and in turn rewrite the diameter as 2•radius, so 2πr. Having rewrote the circumference as 2πr, we just multiply this by r to derive 2πr^2.

As you can see, this attempt of a proof involves only three steps. It is short, simple, intuitive, and, dare I say, eloquent. Above all else, however, it is also wrong. You can even see this visually, just by eyeballing the image.

Now, I need no convincing on what the area of a circle actually is. I've looked up actual proofs online showing why it is πr^2, and I also simply trust the likes of Archimedes, as well as Pythagoras, Newton, the team at NASA, etc. What I need convincing of, however, is that my proof is incorrect. By "convincing", I don't mean it in the usual sense, but in quite a literal one. As in, I can consciously accept that my attempted proof is incorrect (again you can even see this visually by comparing the areas of the circle/rectangle), but my heart and soul cannot, because I've managed to construct such a simple, easy to follow, and intuitive proof that ended up being false. It's clear that the mistake made along the way was not a technical one. I mean sure, the mistake is that my derived equation is off by a factor of exactly 2, but there's something very fundamental about the nature of maths itself that I clearly have not grasped, and I have absolutely no starting point to work from in trying to figure out what that is. This is, to me, like trying to understand why two plus two is two instead of four, for I cannot wrap my head around it.

Answer 1

This argument. fails (as you know) because you can't just think of the circle as a collection of radii, then move the radii into a rectangle to get the area. You are trying to add up the areas of infinitely many segments each of which has $0$ area - it's no surprise that you get conflicting answers.

But I congratulate you on your clever attempt. Here's how to make it work. Instead of cutting the circle into infinitely many radii, cut into pie slices and think about what happens as they get thinner and thinner (but always real slices).

The picture is from https://www.colorado.edu/csl/2017/03/23/slices-pi .

Answer 2

You have exploited the fact the each radius is infinitely thin. Taking something finite, cutting it up into an infinite number of infinitely small things, and then reassembling them does not preserve area. There are lots of paradoxes like yours based on this fact. Check out the Banach-Tarski paradox for one of the most mind blowing. (Although actually, the B.T. paradox only breaks things into two pieces, but I'd argue that they're infinitely complicated. But it is another case of area (okay, volume) not being preserved when cutting up sets.)

If you want to get a visceral feel for what went wrong, try reversing your process: cut out that strip of paper, and start packing it in to your circle. You'll notice that it starts to get all bunched up, with the bunching getting worse the closer you get to the center.

The proper way to add up an infinite number of infinitely small things is with integrals. What your two different ways of slicing up the area amounts to is using two different coordinate systems, and when changing coordinates there is a scaling factor, the Jacobian, that you multiply by that exactly compensates for the stretching or bunching you've noticed.

Answer 3

Consider using your strategy to compute the area of a 1x1 square: Place the square on the $x$-axis between 0 and 1, divide it into uncountably many vertical line segments, and move each segment horizontally from its initial position $x$ to position $5x$. The moved segments fill a 5x1 rectangle, so the area of the square must be 5.

Of course this is wrong, because when you move the segments, you're stretching the shape in a way that doesn't preserve its area.