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 (x0, y0, z0) and radius r is the set of all points (x,y,z) such that
- (x - x0)2 + (y - y0)2 + (z - z0)2 = r2
- 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 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.
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.