steradians

The steradian is the SI unit of solid angle, a measure of how large a directionally localized patch of the sphere around a point appears. It is the three-dimensional analogue of the radian.

For a sphere of radius r, the solid angle Ω subtended by a surface patch A on the sphere is Ω = A / r^2. The total solid angle around a point is 4π steradians, equal to the surface area of a unit sphere.

Although defined as a ratio of areas, the steradian is treated as a unit in the SI, with 1 sr equal to 1 m^2 / m^2. In practice it is used to quantify solid angle rather than as a simple pure number.

For an infinitesimal surface element dA at distance r, the differential solid angle is dΩ = dA / r^2. A common exact form is the solid angle of a cone with half-angle θ: Ω = 2π(1 − cos θ). A hemisphere corresponds to 2π steradians.

Steradians are used in radiometry and photometry to describe radiant flux or irradiance through a surface, in astronomy to express sky coverage, and in computer graphics for lighting calculations and field of view. They provide a precise way to weight contributions by angular extent and are fundamental to relating directional measurements to surface areas on spheres.