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?

I'll add a Chinese version for fun.
The ancient Chinese had another way to calculate this volume.

The principle is same as Cavalieri's Principle; the difference is using the intersection of two perpendicular cylinders, a "bicylinder", to pack the sphere. The Chinese name for this shape is 牟合方蓋 or "mouhefanggai" (meaning two square umbrellas).
Every plane (parallel to the cylinders' axes) intersects in a square with the bicylinder and intersects in a circle with the sphere (for more pictures and an animation, see http://phdfishman.blogspot.com/2010/02/blog-post_07.html ). This means that the volume ratio of the sphere to the bicylinder is proportional to the areas of the circles and squares: $\dfrac{\pi r^2}{(2r)^2}=\dfrac{\pi}{4}$.
Now the question becomes calculating the volume of the bicylinder (white). It is also very difficult, so add a cube (red) packing the bicylinder (white). Now when the plane intersects the cube, it forms another larger square. The extra area in the large square (the big square from the cube minus the smaller square from the bicylinder), is the same as $4$ small squares (blue). As the plane cutting through the solids moves, these blue squares will form $4$ small pyramids in the corners of the cube with isosceles triangle sides and their apex at the edge of the cube. Moving through the whole bicylinder generates a total of $8$ pyramids.
Now we can calculate the volume of the cube (red) minus the volume of the eight pyramids (blue) to get the volume of the bicylinder (white). The volume of the pyramids is:$$8\cdot \frac{1}{3}r^2\cdot r=\frac{8}{3}r^3,$$and then we can calculate that the bicylinder volume is $(2r)^3-\dfrac{8}{3}r^3=\dfrac{16}{3}r^3$.
Finally, using the ratio of the volumes of the bicylinder and the sphere from above, the sphere's volume is $\dfrac{\pi}{4}\dfrac{16}{3}r^3=\dfrac{4\pi}{3}r^3$.
Now you can see that $3$ is from the pyramids and $4$ is from the cube! They are not random.

This picture shows the geometric relationships.
