Steradian: the SI unit of solid angle

Definition and basic relations

By definition, a solid angle Ω (in steradians) subtended at the center of a sphere of radius r equals the area A of the portion of the sphere's surface divided by r²: Ω = A / r². Because both A and r² carry units of area, the steradian is a dimensionless derived unit, though it is customarily written as "sr".

• The total solid angle around a point (the whole sphere) is 4π sr.
• A hemisphere spans 2π sr; a small cone of half‑angle θ has Ω = 2π(1 − cos θ), which for small θ approximates Ω ≈ πθ².
• One steradian is about 1/(4π) of a full sphere — roughly one twelfth of the sphere's surface area — and equals about 3,282.8 square degrees.

Uses and examples

Steradians appear wherever angular extent in three dimensions matters. In photometry and radiometry they relate directional flux and intensity: luminous intensity (candela) is luminous flux (lumens) per steradian, and radiant intensity has units of watts per steradian. In astronomy and remote sensing solid angle measures apparent sizes of extended sources, while in antenna theory it helps quantify beam widths and directivity. Practical examples include estimating how much of the sky a telescope field of view covers or the fraction of emitted power falling on a detector.

Properties, notation and remarks

Although the steradian is treated as a named SI unit (see SI derived unit), it is formally dimensionless because it is a ratio of areas. That can sometimes lead to omission of the unit in algebraic expressions; nonetheless writing "sr" clarifies angular quantities and prevents confusion with plain area or squared angle units. The steradian specifically characterizes three-dimensional angular extent, in contrast to the one‑dimensional radian, which measures planar angles.

Common pitfalls include interpreting steradians as literal surface areas (they measure angular extent independent of the sphere's radius) and confusing numeric small‑angle approximations with exact formulas. When communicating measurements of directional quantities, including the unit "sr" makes the physical meaning explicit and aids comparison across disciplines.