A **sphere** is, roughly speaking, a ball-shaped object. In mathematics, a sphere consists only of a surface and is therefore hollow. In non-mathematical usage a sphere is often considered to be solid.

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.

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

The surface area of a sphere of radius *r* is 4π*r*^{2},
and its volume is 4π*r*^{3}/3.