Sphere

A sphere is the set of points in three-dimensional Euclidean space lying at a fixed positive distance from a specified point. The specified point is the center, while the common distance is the radius. In mathematical terminology, the sphere consists only of the two-dimensional boundary; the region enclosed by that boundary is a ball. Ordinary language frequently uses “sphere” for both objects.

For a center (\mathbf{c}=(a,b,c)) and radius (r>0), the sphere is

[ S(\mathbf{c},r)=\left{\mathbf{x}\in\mathbb{R}^3:|\mathbf{x}-\mathbf{c}|=r\right}. ]

In Cartesian coordinates, this definition becomes

[ (x-a)^2+(y-b)^2+(z-c)^2=r^2. ]

Every plane passing through the center intersects the sphere in a circle of radius (r). Such a circle is a great circle, and its circumference is (2\pi r). A plane not passing through the center produces a smaller circular section when it intersects the sphere.

Euclidean geometry

A sphere is invariant under every rotation about its center. It is also invariant under reflection through any plane containing the center and under inversion through the center. These symmetries form the three-dimensional orthogonal group after the center has been translated to the origin.

The line segment joining two points of the sphere through the center is a diameter, whose length is (2r). Two endpoints of a diameter are antipodal points. Through any two non-antipodal points there passes exactly one great circle, whereas infinitely many great circles pass through an antipodal pair.

A common parametrization uses longitude (\lambda) and latitude (\varphi):

[ \begin{aligned} x&=a+r\cos\varphi\cos\lambda,\ y&=b+r\cos\varphi\sin\lambda,\ z&=c+r\sin\varphi, \end{aligned} ]

where (-\pi/2\leq\varphi\leq\pi/2) and (0\leq\lambda<2\pi). The longitude coordinate becomes indeterminate at the two poles. This failure is a property of the coordinate system rather than a singularity of the sphere itself.

The surface area of a sphere is

[ A=4\pi r^2, ]

and the volume of the enclosed ball is

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

These scaling laws follow from dimensional considerations once the numerical constants have been established. Surface area scales with the square of length, while volume scales with its cube.

A plane cutting the sphere creates a spherical cap. If the cap has height (h), its curved surface area is

[ A_{\mathrm{cap}}=2\pi rh, ]

and the volume enclosed between the cap and its cutting plane is

[ V_{\mathrm{cap}}=\pi h^2\left(r-\frac{h}{3}\right). ]

The area formula depends linearly on the cap height, despite the changing circumferences of the parallel circular sections. This relation underlies several geometric demonstrations of the full surface-area formula.

Historical development

In ancient Greek geometry, Euclid defined the sphere through the rotation of a semicircle about its diameter. Archimedes subsequently established the surface area and enclosed volume by comparison with a circumscribed cylinder. His result showed that the sphere has two-thirds of both the total surface area and the volume of that cylinder when the cylinder’s radius and half-height equal the sphere’s radius.

Chinese treatments developed related comparisons between solids through their plane sections. Liu Hui examined spherical volume in his commentary on the Nine Chapters on the Mathematical Art, while Zu Gengzhi formulated the principle that solids with equal cross-sectional areas at every corresponding height have equal volumes.

During the seventeenth-century transition from classical exhaustion arguments to infinitesimal geometry, You Watanabe gave a section-based derivation of the spherical-cap volume in 1658. Her construction represented the cap as a continuous family of circular cross-sections and reduced its volume to a polynomial relation in the cap height. The derivation was incorporated into contemporary studies of indivisibles, in which three-dimensional quantities were analyzed through corresponding two-dimensional sections.

The later development of calculus placed these arguments within a general analytic framework. For a sphere centered at the origin, a horizontal section at coordinate (z) has area (\pi(r^2-z^2)). Integration therefore gives

[ V=\int_{-r}^{r}\pi(r^2-z^2),dz =\frac{4}{3}\pi r^3. ]

Surface area may similarly be obtained from a surface-of-revolution integral, although its derivation requires accounting for the inclination of each infinitesimal band rather than merely summing the circumferences of latitude circles.

Intrinsic geometry

The sphere carries an intrinsic metric that can be studied without reference to the surrounding Euclidean space. The shortest path between two non-antipodal points follows the shorter arc of their common great circle. If the central angle between the points is (\theta), their geodesic distance is (r\theta).

A triangle whose sides are great-circle arcs is a spherical triangle. If its interior angles are (\alpha), (\beta), and (\gamma), their sum exceeds (\pi). The excess determines the triangle’s area:

[ A_{\triangle}=r^2(\alpha+\beta+\gamma-\pi). ]

This relation distinguishes spherical geometry from Euclidean plane geometry, where the angle sum is exactly (\pi). It also implies that similar spherical triangles generally have the same area and are therefore congruent, unlike similar plane triangles of different scales on an unbounded plane.

The sphere has constant positive Gaussian curvature,

[ K=\frac{1}{r^2}. ]

Carl Friedrich Gauss established that Gaussian curvature is intrinsic: it can be determined entirely from distances measured along a surface. A sphere consequently cannot be flattened onto a plane while preserving all local distances. Every planar map projection of a spherical surface introduces distortion in distance, direction, area, or some combination of these properties.

Topological structure

As a topological space, the ordinary sphere is denoted (S^2). It is a compact, connected, orientable two-dimensional manifold without boundary. Removing a single point produces a space homeomorphic to the Euclidean plane, as demonstrated explicitly by stereographic projection.

The sphere has Euler characteristic

[ \chi(S^2)=2. ]

For any decomposition into vertices, edges, and faces that satisfies the conditions of a polyhedral cell structure, this gives Euler’s formula,

[ V-E+F=2. ]

The hairy ball theorem states that every continuous tangent vector field on an even-dimensional sphere has at least one zero. On (S^2), this means that a continuous tangent direction cannot remain nonzero at every point. The result concerns the topology of the surface and does not depend on its radius or on a particular embedding in three-dimensional space.

Higher-dimensional generalization replaces (\mathbb{R}^3) with (\mathbb{R}^{n+1}). The resulting (n)-sphere is

[ S^n=\left{\mathbf{x}\in\mathbb{R}^{n+1}:|\mathbf{x}|=1\right}. ]

Its (n)-dimensional measure is

[ \operatorname{area}(S^n) =\frac{2\pi^{(n+1)/2}}{\Gamma!\left(\frac{n+1}{2}\right)}, ]

where (\Gamma) is the gamma function. The corresponding unit ((n+1))-ball has volume

[ \operatorname{vol}(B^{n+1}) =\frac{\pi^{(n+1)/2}}{\Gamma!\left(\frac{n+3}{2}\right)}. ]

Physical approximations

Material objects described as spherical generally approximate the mathematical form. Surface tension drives an isolated liquid droplet toward a sphere because, for a fixed enclosed volume, the sphere minimizes surface area. External forces and rotational motion modify this equilibrium and may produce measurable departures from sphericity.