Volume of a Sphere
The volume of a sphere is four-thirds times $\pi$ times the radius cubed.
Formula
V = \frac{4}{3}\pi r^3
Definition
The volume of a sphere (a ball shape) with radius $r$ is $V = (4/3)\pi r^3$: cube the radius, then multiply by $\pi$ and $4/3$. Volume scales as $r^3$, so doubling the radius multiplies volume by $8$; a sphere and its circumscribed cylinder (radius $r$, height $2r$) have volumes in ratio $2:3$, and hemisphere volume is $(2/3)\pi r^3$. By integration, $$V = \int_{-r}^{r} \pi(r^2 - z^2) \, dz = \pi\left[r^2 z - \frac{z^3}{3}\right]_{-r}^{r} = \frac{4}{3}\pi r^3,$$ or in spherical coordinates, $$V = \int_0^{2\pi}\int_0^\pi\int_0^r \rho^2\sin\varphi \, d\rho \, d\varphi \, d\theta = \frac{4}{3}\pi r^3.$$
Example
A ball with radius $3$ cm has $V = (4/3) \times \pi \times 27 = 36\pi = 113.1$ cm$^3$, and a basketball with radius $12$ cm has $V = (4/3)\pi \cdot 1728 = 2304\pi = 7238$ cm$^3$. Earth (radius $6371$ km) has $V = (4/3)\pi(6371)^3 = 1.083 \times 10^{12}$ km$^3$, while Jupiter, with radius $69{,}911$ km, has $V_{Jupiter} / V_{Earth} = (69911/6371)^3 = 1321$ times Earth's volume. The derivative of $V = (4/3)\pi r^3$ with respect to $r$ is $4\pi r^2 = SA$, the surface area, since a thin shell of thickness $dr$ at radius $r$ contributes $SA \cdot dr$ to the volume.
Key Insight
A sphere fits perfectly inside a cylinder with the same diameter and height, taking up exactly $2/3$ of that cylinder's volume, a result Archimedes discovered and was so delighted by that he asked for it to be carved on his tombstone. Volume scales as the cube of linear dimensions, so a star $10$ times Earth's radius has $1000$ times Earth's volume, explaining why large planets and stars accumulate mass so rapidly as they grow. The relationship $V'(r) = SA(r)$ generalizes to any dimension: the n-dimensional volume of an n-ball is the integral of its $(n-1)$-dimensional surface area, a beautiful unification of measurement in all dimensions.