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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 (x0y0z0) and radius r is the set of all points (x,y,z) such that

(x - x0)2 + (y - y0)2 + (z - z0)2 = r2

The points on the sphere with radius 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)
(see trigonometric functions and spherical coordinates).

The surface area of a sphere of radius r is 4πr2, and its volume is 4πr3/3. 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 Sn and is often referred to as "the" n-sphere.

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