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 am no where near as proficient in math as any of the people who answered this before me, but nonetheless I would like to add a simplified version;
A cylinder's volume is: 
$$\pi r^2h$$
A cone's volume is $\frac{1}{3}$ that of a cylinder of equal height and radius:
$$\frac{1}{3}\pi r^2h$$
A sphere's volume is two cones, each of equal height and radius to that of the sphere's: 
$$\frac{1}{3}\pi r^2h + \frac{1}{3}\pi r^2h$$
The height of the sphere is equal to it's diameter $(r + r)$ so the earlier equation can be rewritten as;
$$\frac{1}{3}\pi r^2(r + r) + \frac{1}{3}\pi r^2(r + r)$$
If we simplify it;
$$\frac{1}{3}\pi r^2(2r) + \frac{1}{3}\pi r^2(2r)$$
Following the math convention of numbers before letters it changes to:
$$\frac{1}{3}2\pi r^2r + \frac{1}{3}2\pi r^2r$$
Combining like terms;
$$r^2\cdot r= r^3$$
and
$$\frac{1}{3}\cdot 2 = \frac{2}{3}$$
The equation now becomes
$$\frac{2}{3}\pi r^3 + \frac{2}{3}\pi r^3$$
Again add the like terms, being the $\frac{2}{3}$ together;
$$\frac{2}{3} + \frac{2}{3} = \frac{4}{3}$$
Finally we get to how $\frac{4}{3}$ is part of the equation;
$$\frac{4}{3}\pi r^3$$
