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SI derived unit of solid angle
A graphical representation of 1 steradian.The sphere has radius r, and in this case the area A of the highlighted surface patch is r2. The solid angle ? equals [ which is in this example. The entire sphere has a solid angle of 4? sr.
The steradian (symbol: sr) or square radian is the SI unit of solid angle. It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radians, projected onto a circle, gives a length on the circumference, a solid angle in steradians, projected onto a sphere, gives an area on the surface. The name is derived from the Greek?stereos 'solid' + radian.
The steradian, like the radian, is a dimensionless unit, the quotient of the area subtended and the square of its distance from the center. Both the numerator and denominator of this ratio have dimension length squared (i.e. , dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different nature, 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 center of a unit sphere by a unit area on its surface. For a general sphere of radiusr, any portion of its surface with area subtends one steradian at its center.
The solid angle is related to the area it cuts out of a sphere:
Because the surface area A of a sphere is 4?r2, the definition implies that a sphere subtends 4? steradians (? 12.56637 sr) at its center. By the same argument, the maximum solid angle that can be subtended at any point is 4? 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 = 2?rh) (where h stands for the "height" of the cap) and the relationship h/r = 1/2? 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 2?, with ? given by:
This angle corresponds to the plane aperture angle of 2? ? 1.144 rad or 65.54°.