Sphere
A sphere is a perfectly round 3-D solid where every point on the surface is the same distance from the center.
Formula
V = \frac{4}{3}\pi r^3; \ SA = 4\pi r^2
Definition
A sphere is a perfectly round 3-D shape, like a ball, where every point on the surface is exactly the same distance (the radius) from the center: $S^2 = \{x \in \mathbb{R}^3 : |x| = r\}$. Volume $V = (4/3)\pi r^3$ and surface area $SA = 4\pi r^2$; the great circle (intersection with a plane through the center) has radius $r$, and a hemisphere is exactly half a sphere. Surface area is computed via the surface integral in spherical coordinates, $$SA = \int_0^\pi\int_0^{2\pi} r^2 \sin\varphi \, d\theta \, d\varphi,$$ and volume follows similarly by integrating in spherical coordinates.
Example
A basketball with radius $12$ cm has volume $(4/3) \times \pi \times 12^3 = 2304\pi = 7238$ cm$^3$ and surface area $4 \times \pi \times 144 = 576\pi = 1810$ cm$^2$. Earth's radius is approximately $6371$ km, giving surface area $4\pi(6371)^2 = 5.1 \times 10^8$ km$^2$ and volume $(4/3)\pi(6371)^3 = 1.08 \times 10^{12}$ km$^3$, of which about $71\%$ of the surface is ocean. The n-sphere $S^{n-1}$ in $\mathbb{R}^n$ has volume $V_n = \pi^{n/2} r^n / \Gamma(n/2 + 1)$: $\pi r^2$ for $n=2$ (disk), $(4/3)\pi r^3$ for $n=3$ (ball), $(\pi^2/2)r^4$ for $n=4$, peaking around $n=5$ or $6$ for $r=1$ before decreasing to $0$ as $n \to \infty$.
Key Insight
The sphere is the most "efficient" 3-D shape, enclosing the most volume for its surface area, which is why nature uses spheres in bubbles, raindrops, and planets. Archimedes showed that a sphere fits perfectly inside a cylinder of the same diameter and height, taking up exactly $2/3$ of that cylinder's volume while its surface area equals the cylinder's lateral surface area, a result he was so proud of that he asked for a sphere-in-cylinder to be carved on his tomb. The fact that n-ball volume goes to $0$ as dimension increases means that in high dimensions nearly all the volume is near the surface, a "concentration of measure" phenomenon fundamental to statistics, machine learning, and random matrix theory.