Steradian

In the SI, solid angle is considered to be a dimensionless quantity, the ratio of the area projected onto a surrounding sphere and the square of the sphere's radius. This is the number of square radians in the solid angle. This means that the SI steradian is the number of square radians in a solid angle equal to one square radian, which of course is the number one. It is useful to distinguish between dimensionless quantities of a different kind, such as the radian (in the SI, a ratio of quantities of dimension length), so the symbol sr is used. For example, radiant intensity can be measured in watts per steradian (W⋅sr⁻¹). The steradian was formerly an SI supplementary unit, but this category was abolished in 1995 and the steradian is now considered an SI derived unit.

The name steradian is derived from the Greek στερεός stereos 'solid' + radian.

Definition

A steradian can be defined as the solid angle subtended at the centre of a unit sphere by a unit area (of any shape) on its surface. For a general sphere of radius r, any portion of its surface with area A = r² subtends one steradian at its centre.

A solid angle in the form of a circular cone is related to the area it cuts out of a sphere:

${\displaystyle \mathrm{\Omega }=\frac{A}{r^2}\text{sr}=\frac{2\pi h}{r}\text{sr},}$

where

• Ω is the solid angle
• A is the surface area of the spherical cap, 2πrh,
• r is the radius of the sphere,
• h is the height of the cap, and
• sr is the unit, steradian; sr = rad².

Because the surface area A of a sphere is 4πr², the definition implies that a sphere subtends 4π steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends 1/4π ≈ 0.07958 of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is 4π sr.