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I've gotten this result a couple of times trying to "derive" basic geometric area/volume equations and I thought I had resolved it internally already. However, I was thinking back again to finding the volume of a sphere and can't understand why this is incorrect for the volume of a sphere: $(\pi r)(\pi r^{2}) = \pi^{2}r^{3}$ Here is an image to 'justify' my intuition: Circle Rotation

The image shows a 180 degree rotation. We know the circumference of a circle is $2\pi r$ which we can divide in half for half a rotation. We can then multiply that by the area $\pi r^{2}$ to find the volume. However, of course this is wrong, but why? We know the actual volume to be $\frac{4}{3}\pi r^{3}$. I know that Archimedes was able to derive this geometrically, so I shouldn't need calculus to find this.

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    $\begingroup$ Just asking: Do you know Pappus's theorem on centroids? $\endgroup$ – Andrew D. Hwang Nov 22 '15 at 18:38
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    $\begingroup$ Most points on the disk (in fact, all except for two) traverse an arc that is shorter than $\pi r$ when you rotate it around its diameter. Therefore multiplying everythhing by $\pi r$ overestimates the volume. $\endgroup$ – hmakholm left over Monica Nov 22 '15 at 18:40
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Think of it this way: If you had the area of the circle as $\pi r^2$, by multiplying the same by an arbitrary constant (in this case $r$), would just transform the circle into a cylinder (with height in this case $r$). Now you multiplied it by $\pi$, so you have essentially the volume of the solid you obtained as $\pi^2 r^3$. Now if you equate this to the correct formula for the volume of a sphere with radius $R$, you see that $\pi^2 r^3 = (4/3) \pi R^2$, which simplifies to $r = R \sqrt[3]{4/3 \pi}$. You have essentially found the volume of a sphere with radius = $r\sqrt[3]{4/3 \pi}$ if your initial circle's radius was $r$.

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