Steradian

Steradian (symbol sr) is the SI unit of solid angle. It is defined as the solid angle subtended at the center of a sphere of radius r by a surface area equal to r^2. Equivalently, the steradian is the ratio dA / r^2, where dA is an infinitesimal area on the sphere. Although the steradian is dimensionless, it is treated as a unit to quantify angular measurements in three dimensions.

A full sphere around a point subtends 4π steradians; on a unit sphere the total surface area is 4π, which equals about 12.566 sr. In practical terms, 1 sr is equivalent to approximately 3282.806 square degrees. For small patches, the differential solid angle is dΩ = sinθ dθ dφ in spherical coordinates, and the corresponding patch area satisfies dA = r^2 dΩ.

Steradians are widely used in radiometry and photometry to describe how much of the field of view a source or detector covers. Radiant flux, irradiance, and radiance are often expressed with respect to solid angles; for example, radiant intensity is the power per unit solid angle (W per sr), and the total flux into a solid angle Ω is obtained by integrating I over Ω. The concept also appears in astronomy, computer graphics, and optical design, where angular extent and directional distributions are essential for modeling light propagation.