A graphical representation of two different steradians. The sphere has radius r, and in this case the area A of the highlighted
spherical cap is r2. The solid angle Ω equals A/r2] sr which is 1 sr in this example. The entire sphere has a solid angle of 4π sr.
The steradian is a
dimensionless unit, the quotient of the area subtended and the square of its distance from the centre. Both the numerator and denominator of this ratio have dimension length squared (i.e. L2/L2 = 1, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different kind, such as the radian (a ratio of quantities of dimension length), so the symbol "sr" is used to indicate a solid angle. 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.
Solid angle of countries and other entities relative to the Earth.
A steradian can be defined as the solid angle
subtended at the centre of a
unit sphere by a unit
area on its surface. For a general sphere of
radiusr, any portion of its surface with area A = r2 subtends one steradian at its centre.
The solid angle is related to the area it cuts out of a sphere:
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.
Section of cone (1) and spherical cap (2) that subtend a solid angle of one steradian inside a sphere
If A = r2, it corresponds to the area of a
spherical cap (A = 2πrh, where h is the "height" of the cap) and the relationship holds. Therefore, in this case, one steradian corresponds to the plane (i.e. radian) angle of the cross-section of a simple cone subtending the plane angle 2θ, with θ given by:
This angle corresponds to the plane aperture angle of 2θ ≈ 1.144 rad or 65.54°.