Sphere

A sphere is a 2-dimensional surface produced by finding the set of all points that are an equal distance from another point in 3-dimensional space. A unit sphere is a sphere that ensures all points on the sphere are distance one from the center. Because it is curved, it is often represented embedded in 3-dimensional space. A sphere is the shape of the exterior of a ball. It is the 2-dimensional hypersphere. Spherical geometry does not follow all of Euclid's axioms.

The sphere of radius r and center c is the set of all points that are at the same distance r from point c in 3-dimensional Euclidean space:

$S=\left\{x\in \mathbb {R} ^{3}|d(x,c)=r\right\}$

where $d(x,c)$ is euclidean metric for three-dimensional space:

$d(x,c)={\sqrt {\sum _{i=1}^{3}(x_{i}-c_{i})^{2}}}$ .

The volume V inside a sphere of radius r can be calculated using expression:

$V=\frac{4}{3}\pi r^{3}$ .

The surface area of a sphere is:

$S = 4 \pi r^2$ .

Generalizations[]

Generalization of sphere for euclidean spaces of arbitrary dimension is hypersphere (n-sphere).

Speaking even more generally, for a metric space (M,d), where M is a set and d is a metric (distance function), the sphere S of radius r and center c is the set:

$S=\{x\in M|d(x,c)=r\}$ .

See Also[]

Dimensionality	Negative One	Zero	One	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	...	Aleph null
Hyperbolic space $\mathbb H^{n}$	—	—	—	Hyperbolic plane $\mathbb H^{2}$	Hyperbolic realm $\mathbb H^{3}$	Hyperbolic flune $\mathbb H^{4}$	Hyperbolic pentrealm $\mathbb H^{5}$	Hyperbolic hexealm $\mathbb H^{6}$	Hyperbolic heptealm $\mathbb H^{7}$	Hyperbolic octealm $\mathbb H^{8}$	Hyperbolic ennealm $\mathbb H^{9}$	Hyperbolic decealm $\mathbb H^{10}$	...	Hyperbolic omegealm $\mathbb H^{\aleph_0}$
Euclidean space $\R^n$	Null polytope $\emptyset$	Point $\mathbb R^{0}$	Euclidean line $\mathbb R^{1}$	Euclidean plane $\mathbb R^{2}$	Euclidean realm $\mathbb R^{3}$	Euclidean flune $\mathbb R^{4}$	Euclidean pentrealm $\mathbb R^{5}$	Euclidean hexealm $\mathbb R^{6}$	Euclidean heptealm $\mathbb R^{7}$	Euclidean octealm $\mathbb R^{8}$	Euclidean ennealm $\mathbb R^{9}$	Euclidean decealm $\mathbb R^{10}$	...	Euclidean omegealm $\mathbb R^{\aleph_0}$
Hypersphere $\mathbb S^{n}$	Null polytope $\emptyset$	Point pair $\mathbb S^{0}$	Circle $\mathbb S^{1}$	Sphere $\mathbb S^{2}$	Glome $\mathbb S^{3}$	Tetrasphere $\mathbb S^{4}$	Pentasphere $\mathbb S^{5}$	Hexasphere $\mathbb S^{6}$	Heptasphere $\mathbb S^{7}$	Octasphere $\mathbb S^{8}$	Enneasphere $\mathbb S^{9}$	Dekasphere $\mathbb S^{10}$	...	Omegasphere $\mathbb S^{\aleph_0}$


$\mathbb S^{2}$	$\mathbb{T}^2$	$2\mathbb{T}^2$	$3\mathbb{T}^2$	$4\mathbb{T}^2$
Sphere	Torus	Double torus	Triple torus	Quadruple torus


$\mathbb S^{2}$	$\mathbb{RP}^2$	$2\mathbb{RP}^2$	$3\mathbb{RP}^2$
Sphere	Real projective plane	Klein bottle	Dyck's surface