Sphere
Geometry & MeasurementA 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. Every point on the surface of a sphere is exactly the same distance (the radius) from the center.
Example
A basketball with radius $12$ cm: Volume $= (4/3) \times \pi \times 12^3 = (4/3) \times \pi \times 1728 = 2304\pi = 7238$ cm$^3$. Surface area $= 4 \times \pi \times 144 = 576\pi = 1810$ cm$^2$.
Key Insight
The sphere is the most "efficient" 3-D shape: it encloses the most volume for its surface area. Nature uses spheres in bubbles, raindrops, and planets for exactly this reason.
Definition
A sphere of radius $r$ is the set of all points in 3-D space at distance $r$ from the center. Volume $V = (4/3)\pi r^3$; Surface area $SA = 4\pi r^2$. The great circle (intersection of the sphere with a plane through the center) has radius $r$. A hemisphere is exactly half a sphere.
Example
Earth's radius is approximately $6371$ km. Surface area $= 4\pi(6371)^2 = 5.1 \times 10^8$ km$^2$. Volume $= (4/3)\pi(6371)^3 = 1.08 \times 10^{12}$ km$^3$. About $71\%$ of the surface area is covered by ocean.
Key Insight
Archimedes showed that a sphere fits perfectly inside a cylinder of the same diameter and height. The sphere's volume is exactly $2/3$ of that cylinder, and the sphere's surface area equals the cylinder's lateral surface area. He was so proud of this result he asked for a sphere-in-cylinder carved on his tomb.
Definition
The sphere $S^2$ of radius $r$ in $\mathbb{R}^3$ is the level set $\{x \in \mathbb{R}^3 : |x| = r\}$. Surface area $SA = 4\pi r^2$ is computed via the surface integral using spherical coordinates: $$SA = \int_0^\pi\int_0^{2\pi} r^2 \sin\varphi \, d\theta \, d\varphi.$$ Volume $V = (4/3)\pi r^3$ follows from integrating the sphere in spherical coordinates.
Example
The n-sphere $S^{n-1}$ in $\mathbb{R}^n$ has volume $V_n = \pi^{n/2} r^n / \Gamma(n/2 + 1)$. For $n=2$ (disk): $\pi r^2$; for $n=3$ (ball): $(4/3)\pi r^3$; for $n=4$: $(\pi^2/2)r^4$. Volume peaks at $n=5$ or $6$ for $r=1$ and decreases to $0$ as $n \to \infty$.
Key Insight
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. This "concentration of measure" phenomenon is fundamental to statistics, machine learning, and the theory of random matrices.