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I was wondering if it was ever possible, in certain cases, for the area of a circle to be equal to the circumference of the same circle squared? If so, how would one come to this derivation? Thanks in advance!

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    $\begingroup$ You want $(2\pi r)^2=\pi r^2$. But that is not possible unless $r=0$. $\endgroup$ – Anurag A Apr 10 '16 at 2:35
  • $\begingroup$ In Euclidean space? I don't think that's possible in that case. $\endgroup$ – velut luna Apr 10 '16 at 2:35
  • $\begingroup$ If you think about it, the circumference squared would mean a square where the two sides were the length of the circumference...since this is necessarily larger than the diameter of the circle, the circle would certainly fit fully inside of the resulting square. $\endgroup$ – Jared Apr 10 '16 at 2:40
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\begin{align} \text{area} & = \pi r^2 \\[10pt] \text{circumference} & = 2\pi r \\[10pt] \frac{\text{circumference}}{2\pi} & = r \\[10pt] \left(\frac{\text{circumference}}{2\pi} \right)^2 & = r^2 \\[10pt] \pi \left(\frac{\text{circumference}}{2\pi} \right)^2 & = \pi r^2 = \text{area} \\[10pt] \frac{\text{cicumference}^2}{4\pi} & = \text{area} \end{align}

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I think not. Here's why:

$$A = \pi r^2$$ $$C = 2 \pi r$$

Now take

$$A = C^2$$ $$\implies \pi r^2 = (2 \pi r)^2 = 4 \pi^2 r^2$$ $$\implies \pi = 4 \pi^2 $$ $$\implies \frac{1}{4} = \pi $$

A more convincing argument would solve for the radius: $$\pi r^2 = 4 \pi^2 r^2$$ $$\implies \pi r^2 - 4 \pi^2 r^2 = r^2(\pi - 4\pi^2) = 0$$ $$\implies r = 0$$

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I will restate my comment: if you think about it, the circumference squared would mean a square where the two sides were the length of the circumference. Since this is necessarily larger than the diameter of the circle, the circle would certainly fit fully inside of the resulting square.

Here is a picture showing that there is no way for the two areas to be equal. The square created from "squaring" the circumference is much much bigger than the original circle (I show 9 circles fitting inside--but there is a lot of unused area--it should be exactly $4\pi$ times the original area of the circle--so there are about 3 and a half more circles that could fit in that area).

enter image description here

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