A **sphere** is, roughly speaking, a ball-shaped object. In mathematics, a sphere is a quadric consisting only of a surface and is therefore hollow.
In non-mathematical usage a sphere is often considered to be solid; mathematicians call this the *interior* of the sphere.

More precisely, a sphere is the set of points in 3-dimensional Euclidean space which are at distance *r* from a fixed point of that space, where *r* is a positive real number called the **radius** of the sphere.

In coordinate geometry a sphere with centre (*x*_{0}, *y*_{0}, *z*_{0}) and radius *r* is the set of all points (*x*,*y*,*z*) such that

- (
*x*-*x*_{0})^{2}+ (*y*-*y*_{0})^{2}+ (*z*-*z*_{0})^{2}=*r*^{2}

*r*and center at the origin can be parametrized via

*x*=*r*cos(φ) sin(θ)*y*=*r*sin(φ) sin(θ) (0 ≤ θ < π and -π < φ ≤ π)*z*=*r*cos(θ)

The surface area of a sphere of radius *r* is 4π*r*^{2},
and its volume is 4π*r*^{3}/3. The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and water drops (in the absence of gravity) are spheres because the surface tension tries to minimize surface area.

The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere. This fact, along with the volume and surface formulas given above, was already known to Archimedes.

A sphere can also be defined as the surface formed by rotating a circle about its diameter. If the circle is replaced by an ellipse, the shape becomes a spheroid.

Spheres can be generalized to other dimensions. For any natural number *n*, an *n*-sphere is the set of points in (*n*+1)-dimensional Euclidean space which are at distance *r* from a fixed point of that space, where *r* is, as before, a positive real number.
A 2-sphere is therefore an ordinary sphere, while a 1-sphere is a circle and a 0-sphere is a pair of points. The *n*-sphere of unit radius centered at the origin is denoted *S*^{n} and is often referred to as "the" *n*-sphere.

An *n*-sphere is an example of a compact *n*-manifold.

## See also

**Sphere Books** was a British paperback publisher of the 1960s - 1980s.

** Sphere** is the name of a book written by Michael Crichton, which was subsequently turned into a movie by the same name.