Sphere

A sphere is the set of all points in three-dimensional Euclidean space that are at a fixed distance, called the radius, from a given point, the center. The distance between the center and any point on the surface is the radius; the diameter is twice the radius. The surface of a sphere is a two-dimensional manifold, while the solid region enclosed by the surface is called a ball or a solid sphere, depending on context.

A sphere can be described algebraically by its center and radius. A sphere of radius r centered at (a, b, c) satisfies (x − a)² + (y − b)² + (z − c)² = r². In particular, a sphere centered at the origin has x² + y² + z² = r². Planes intersect a sphere in circles; a line intersects at most in two points; a tangent plane touches at exactly one point.

Key measurements include the surface area and the volume of the solid ball. The surface area of a sphere with radius r is 4πr², and the volume of the enclosed ball is (4/3)πr³. A unit sphere has radius 1. The sphere also features in various geometric constructions, such as inscribed and circumscribed spheres around polyhedra.

Related concepts include great circles, which are the largest circles on a sphere and correspond to the geodesics under the standard spherical geometry. Spherical coordinates (r, θ, φ) describe points on a sphere of fixed radius r. In applications, spheres model celestial bodies to a good approximation, enable computer graphics shading and rendering, and arise in physics and engineering analyses.