<s>[INST] Why is the volume of a sphere $\frac{4}{3}\pi r^3$? I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! How could somebody guess something like this for the formula?
 [/INST] Let us consider the following sphere of radius r with an inscribed pyramid with base area G:

We can calculate the volume of the inscribed pyramid as follows:
$V_{pyramid} = \frac{1}{3} \cdot G \cdot r$
Notice that we can now express $V_{sphere}$ as the sum of the volume of infinitely small pyramids over the total surface area of the sphere, whereby $A_{sphere} = 4 \cdot \pi \cdot r^2$. When doing so, we get the following result:
$V_{sphere} = \sum volume\:of\:pyramid \\
= \sum \frac{1}{3} \cdot G \cdot r \\
= \sum \frac{1}{3} \cdot (4\pi r^2) \cdot r \\
= \frac{4}{3} \pi r^3$
So what do we take from this? The factor $\frac{4}{3}$ can be intuitively considered as a composition of the factor $4$ deriving from the area of the sphere and the factor $\frac{1}{3}$ coming from the volume of a pyramid.
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