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I am trying to understand the area of a circle's formula $\pi r^2$ from a different perspective. I've already understood 2 intuitions: infinite isoceles triangles of height $r$ and area $r*b/2$ where the base $b$ makes up the circumference of the circle. I've also come across the calculus explanation where we stretch out the circumference from radius $r$ all the way to radius 0 forming a right triangle of base $r$ and height $2\pi r$ then we just take the area of that triangle to arrive at the formula $\pi r^2$.

I'm now trying to look at it in 2 different ways:

1) $({r \over 2})2\pi r$ : which would mean to add the circumference of the circle $2\pi r$, $r/2$ times.

or

2) $r\pi r$ : which could be understood as adding together $r$ semi-circles $\pi r$.

I tried to represent assumption 1) graphically and drew this image:

radius 4

The idea I came up with was that if the area of a circle is the circumference added to itself half the radius times $(r/2)2\pi r$ then that meant that I could break a circle up into parts that combined would make these 2 circumferences.

So given a circle of radius 4, I tried combining the circumference of radius 1 to the one of radius 3 and that left me with 2 circumferences of radius $2\pi r$ and one of radius 2 and circumference $\pi r$. For $r = 4$:

$$\pi r^2 + \pi r = 2\pi r + \frac{3\pi r}{2} + \frac{\pi r}{2} + \pi r$$

That equation is obviously not the area of a circle. The correct one would be:

$$\pi r^2 = 2\pi r + \frac{3\pi r}{2} + \frac{\pi r}{2}$$

But then i would just be forgetting the radius 2 circumference $\pi r$. I'm probably forgetting something, but I don't understand why the area of a circle does not account for that middle circumference. Here's a generalization I created that works for finding the area of a circle following the intuition I presented:

$$\pi r^2 = \sum_{i=1}^\frac{r}{2} \frac{2\pi ri}{r} + \frac{2\pi r (r-i)}{r} = \sum_{i=1}^\frac{r}{2} 2\pi i + 2\pi(r-i) = \sum_{i=1}^\frac{r}{2} 2\pi (i + (r-i))$$

This summation formula works like a charm but I dont understand where the "red circle" of radius 2 went, since I just got two $2\pi r$.

Does anyone know what went wrong in my reasoning? What am I missing? Why is my intuition telling me the area of a circle of radius 4 is $\pi r^2 + \pi r$ instead of $\pi r^2$?

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  • $\begingroup$ Here is an intuition why $\,\pi r^2 + \pi r\,$ can not possibly represent the area of the circle: it is not dimensionally correct. Otherwise put, if your formula were right, then the ratio between the areas of two circles would be different if you measured the radii in meters vs. centimeters for example. $\endgroup$ – dxiv Aug 2 '17 at 22:01
  • $\begingroup$ So to be clear, you're trying to get an intuition for the area of a circle, given that the circumference of the circle is $2\pi r$? Or are you just asking "why is $\pi$ what it is"? The latter question is sort of philosophical. $\endgroup$ – Elliot G Aug 2 '17 at 22:06
  • $\begingroup$ yea the first, trying to get an intuition for the area of a circle $\endgroup$ – Danilo Souza Morães Aug 4 '17 at 1:41
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Essentially you're dividing your circle into $4$ annuli of width $1$ each.

That's all well and good, but then you seem to be saying that the area of each annulus is $1$ times its length measured along the outside.

But actually the area of a "fat curve" of constant width, is the width of the curve times the length of the curve's mid-line. So you should be adding up the lengths of circles with radius $\frac12$, $1\frac12$, $2\frac12$, and $3\frac12$ instead -- giving you a sum of $8$, which does equal $\frac12 r^2$ in this case.

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  • $\begingroup$ I was looking at the page right when you answered the question but it took me some time to digest it and actually understand the full meaning of what you said. That was brilliant. I cannot thank you enough for helping me have this beautiful insight.I'm really thankful. I'll think about what you said for a few more hours but I can already consider this question answered. Thanks again! $\endgroup$ – Danilo Souza Morães Aug 4 '17 at 3:14

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